Two dimentional transform
Post on 22-May-2015
805 Views
Preview:
DESCRIPTION
Transcript
TransformationAn operation that changes one
configuration into another
Types of TransformationGeometric transformation
Object itself is transformed relative to a stationary co-ordinate
Co-ordinate transformation Co-ordinate system is transformed relative to an
object Object is held stationary
2D Geometric TransformationsA two dimensional transformation is any
operation on a point in space (x y) that maps that points coordinates into a new set of coordinates (x1 y1)
Instead of applying a transformation to every point in every line that makes up an object the transformation is applied only to the vertices of the object and then new lines are drawn between the resulting endpoints
2D Geometric Transformations
Translate
Rotate Scale
Shear
2D TranslationOne of rigid-body transformation which move objects
withoutdeformation
Translate an object by Adding offsets to coordinates to generatenew coordinates positions
Set tx ty be the translation distance we have
Prsquo=P+TTranslation moves the object without
deformation
P
Prsquo
Txtxx ytyy
y
xP
y
x
t
tT
y
xP
Basic 2D TranslationTo move a line segment apply the
transformation equation to each of the two line endpoints and redraw the line between new endpoints
To move a polygon apply the transformation equation to coordinates of each vertex and regenerate the polygon using the new set of vertex coordinates
Example Translate a polygon with coordinates A(25) B(710) and c(102) by 3 units in x direction and 4 units in y direction
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D Geometric TransformationsA two dimensional transformation is any
operation on a point in space (x y) that maps that points coordinates into a new set of coordinates (x1 y1)
Instead of applying a transformation to every point in every line that makes up an object the transformation is applied only to the vertices of the object and then new lines are drawn between the resulting endpoints
2D Geometric Transformations
Translate
Rotate Scale
Shear
2D TranslationOne of rigid-body transformation which move objects
withoutdeformation
Translate an object by Adding offsets to coordinates to generatenew coordinates positions
Set tx ty be the translation distance we have
Prsquo=P+TTranslation moves the object without
deformation
P
Prsquo
Txtxx ytyy
y
xP
y
x
t
tT
y
xP
Basic 2D TranslationTo move a line segment apply the
transformation equation to each of the two line endpoints and redraw the line between new endpoints
To move a polygon apply the transformation equation to coordinates of each vertex and regenerate the polygon using the new set of vertex coordinates
Example Translate a polygon with coordinates A(25) B(710) and c(102) by 3 units in x direction and 4 units in y direction
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D Geometric Transformations
Translate
Rotate Scale
Shear
2D TranslationOne of rigid-body transformation which move objects
withoutdeformation
Translate an object by Adding offsets to coordinates to generatenew coordinates positions
Set tx ty be the translation distance we have
Prsquo=P+TTranslation moves the object without
deformation
P
Prsquo
Txtxx ytyy
y
xP
y
x
t
tT
y
xP
Basic 2D TranslationTo move a line segment apply the
transformation equation to each of the two line endpoints and redraw the line between new endpoints
To move a polygon apply the transformation equation to coordinates of each vertex and regenerate the polygon using the new set of vertex coordinates
Example Translate a polygon with coordinates A(25) B(710) and c(102) by 3 units in x direction and 4 units in y direction
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D TranslationOne of rigid-body transformation which move objects
withoutdeformation
Translate an object by Adding offsets to coordinates to generatenew coordinates positions
Set tx ty be the translation distance we have
Prsquo=P+TTranslation moves the object without
deformation
P
Prsquo
Txtxx ytyy
y
xP
y
x
t
tT
y
xP
Basic 2D TranslationTo move a line segment apply the
transformation equation to each of the two line endpoints and redraw the line between new endpoints
To move a polygon apply the transformation equation to coordinates of each vertex and regenerate the polygon using the new set of vertex coordinates
Example Translate a polygon with coordinates A(25) B(710) and c(102) by 3 units in x direction and 4 units in y direction
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Basic 2D TranslationTo move a line segment apply the
transformation equation to each of the two line endpoints and redraw the line between new endpoints
To move a polygon apply the transformation equation to coordinates of each vertex and regenerate the polygon using the new set of vertex coordinates
Example Translate a polygon with coordinates A(25) B(710) and c(102) by 3 units in x direction and 4 units in y direction
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Example Translate a polygon with coordinates A(25) B(710) and c(102) by 3 units in x direction and 4 units in y direction
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D RotationObject is rotated ϴdeg about the originϴ gt 0 ndash rotation is counter clock wiseϴ lt 0 ndash rotation is clock wise
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2-D Rotationx = r cos ()y = r sin ()xrsquo = r cos ( + )yrsquo = r sin ( + )
Trig Identityhellipxrsquo = r cos() cos() ndash r sin() sin()yrsquo = r sin() sin() + r cos() cos()
Substitutehellipxrsquo = x cos() - y sin()yrsquo = x sin() + y cos()
(x y)
(xrsquo yrsquo)
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Basic 2D Geometric Transformations2D Rotation matrix
Prsquo=RP
cossin
sincosR
ΦΦ
(xy)rr θ
(xrsquoyrsquo)
y
x
y
x
cossin
sincos
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Basic 2D Geometric Transformations2D Rotation
Rotation for a point about any specified position (xr yr)
xrsquo=xr+(x - xr) cos θ ndash (y - yr) sin θ
yrsquo=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each of the endpoints and redrawing the line between the new end points
A polygon is rotated by applying the rotation formula to each of the vertices and redrawing the polygon using new vertex coordinates
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
ExampleA point (43) is rotated counterclockwise by an angle 45deg find the rotation matrix and resultant point
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Basic 2D Geometric Transformations2D Scaling
Scaling is the process of expanding or compressing the dimension of an object
Simple 2D scaling is performed by multiplying object positions (x y) by scaling factors sx and sy
xrsquo = x sx
yrsquo = y sy
or Prsquo = SP
y
x
s
s
y
x
y
x
0
0
P(xy)
Prsquo(xrsquoyrsquo)
xsx x
sy y
y
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D ScalingAny positive value can
be used as scaling factor Sf lt 1 reduce the size of
the objectSf gt 1 enlarge the object
Sf = 1 then the object stays unchanged
If sx = sy we call it uniform scaling
If scaling factor lt1 then the object moves closer to the origin and If scaling factor gt1 then the object moves farther from the origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
1
2
1
3
3
6
3
9
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Basic 2D Geometric Transformations2D Scaling
We can control the location of the scaled object by choosing a position called the fixed point (xf yf)
xrsquo ndash xf = (x ndash xf) sx yrsquo ndash yf = (y ndash yf) sy
xrsquo=x sx + xf (1 ndash sx)
yrsquo=y sy + yf (1 ndash sy)
Polygons are scaled by applying the above formula to each vertex then regenerating the polygon using the transformed vertices
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Example
Scale the polygon with co-ordinates A(25) B(710) and c(102) by 2 units in x direction and 2 units in y direction
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Homogeneous Coordinates
Expand each 2D coordinate (x y) to three element representation (xh yh h) called homogenous coordinates
h is the homogenous parameter such that
x = xhh y = yhh
A convenient choice is to choose h = 1
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Homogeneous Coordinates for translation
2D Translation Matrix
or Prsquo = T(txty)P
1100
10
01
1
y
x
t
t
y
x
y
x
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Homogeneous Coordinates for rotation
2D Rotation Matrix
or Prsquo = R(θ)P
1100
0cossin
0sincos
1
y
x
y
x
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Homogeneous Coordinates for scaling2D Scaling Matrix
or Prsquo = S(sxsy)P
1100
00
00
1
y
x
s
s
y
x
y
x
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Inverse Transformations2D Inverse Translation Matrix
100
10
011
y
x
t
t
T
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Inverse Transformations2D Inverse Rotation Matrix
100
0cossin
0sincos1R
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Inverse Transformations2D Inverse Scaling Matrix
100
01
0
001
1
y
x
s
s
S
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D Composite TransformationsWe can setup a sequence of
transformations as a composite transformation matrix by calculating the product of the individual transformations
Prsquo=M2M1P
Prsquo=MP
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D Composite TransformationsComposite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D Composite TransformationsComposite 2D Rotations
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
2D Composite TransformationsComposite 2D Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
100
sin)cos1(cossin
sin)cos1(sincos
100
10
01
100
0cossin
0sincos
100
10
01
rr
rr
r
r
r
r
xy
yx
y
x
y
x
rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xry
r)(xry
r)(xry
r)(xry
r)
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
100
)1(0
)1(0
100
10
01
100
00
00
100
10
01
yfy
xfx
f
fx
f
f
sys
sxs
y
x
s
s
y
x
y
Translate Scale Translate
(xryr
)(xry
r)(xry
r)(xry
r)
yxffffyxff ssyxSyxTssSyxT
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Another Example
Scale
Translate
Rotate
Translate
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
ExampleI sat in the car and find the side mirror is 04m
onmy right and 03m in my front
bull I started my car and drove 5m forward turned 30
degrees to right moved 5m forward again andturned 45 degrees to the right and stopped
bull What is the position of the side mirror nowrelative to where I was sitting in the beginning
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Other Two Dimensional Transformations
ReflectionTransformation that produces a mirror image of an object
Image is generated relative to an axis of reflection by rotating the object 180deg about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Reflection about the line y=0 (the x axis)
100
010
001
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Reflection about the line x=0 (the y axis)
100
010
001
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Reflection about the origin
100
010
001
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Reflection when x = y
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Example
Consider the triangle ABC with co-ordinates x(41) y(52) z(43) Reflect the triangle about the x axis and then about the line y = -x
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
ShearTransformation that distorts the shape
of an object is called shear transformation
Two shearing transformation usedShift X co-ordinates valuesShift Y co-ordinates values
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
X shear
y
x
(01) (11)
(10)(00)
y
x
(21) (31)
(10)(00)shx=2
100
010
01 xsh
yy
yshxx x
Preserve Y coordinates but change the X coordinates values
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Y shearPreserve X coordinates but change the Y coordinates values
xrsquo = x yrsquo = y + Shy x
y
x
(01) (11)
(10)(00)
y
x
(01)
(12)
(11)(00)
100
01
001
ysh
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Example
Perform x shear and y shear along on a triangle A(21) B(43) C(23) sh = 2
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Shear relative to other axisX shear with reference to Y axis
x
y
1
1
yref = -1
x
shx = frac12 yref = -1
1
1 2 3
yref = -1
( )x refx x sh y y
y y
1
0 1 0
1 0 0 1 1
x x refx sh sh y x
y y
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Shear relative to other axisY shear with reference to X axis
( )y ref
x x
y y sh x x
1 0 0
1
1 0 0 1 1x y ref
x x
y sh sh x y
x
y
1
1xref = -1
y
x
1
1
2
xref = -1
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Example
Apply shearing transformation to square with A(00) B(10) C(11) D(01)
Shear parameter value is 05 relative to line Yref = - 1 and Xref = - 1
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
321321321 M)MM()MM(MMMM
Associative properties
Transformation is not commutative (CopyCD)Order of transformation may affect
transformation position
Matrix Concatenation Properties
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Transformations between 2D Transformations between 2D Coordinate SystemsCoordinate Systems
To translate object descriptions from xy coordinates to xrsquoyrsquo coordinates we set up a transformation that superimposes the xrsquoyrsquo axes onto the xy axes This is done in two steps
1Translate so that the origin (x0 y0) of the xrsquoyrsquo system is moved to the origin (0 0) of the xy system
2Rotate the xrsquo axis onto the x axis
x
y
x0
xrsquo yrsquo
y0
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
Translation(xrsquoyrsquo)=Tv(xy)xrsquo= x ndash txyrsquo=y-ty
Rotation about origin(xrsquoyrsquo)=Rϴ(xy)xrsquo= xcosϴ + ysinϴ
yrsquo= -xsinϴ + ycosϴ
Scaling with origin(xrsquo yrsquo)=Ssx sy(xy)xrsquo= (1sx)xyrsquo= (1sy)y
Reflection about X axis(xrsquoyrsquo)= Mx(xy)xrsquo= xyrsquo= -y
Reflection about Y axis(xrsquoyrsquo)= My(xy)xrsquo= -xyrsquo= y
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
ExampleFind the xrsquoyrsquo-coordinates of the xy points (10
20) and (35 20) as shown in the figure below
x
y
30
xrsquo
yrsquo
1030ordm
(10 20)
(35 20)
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
ExampleFind the xrsquoyrsquo-coordinates of the rectangle
shown in the figure below
x
y
10
xrsquo
yrsquo 10
60ordm
20
top related