Transcript
© 2005 Pearson Education
Computer Graphics
© 2005 Pearson Education
Chapter 5Geometric
Transformations
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Geometric Transformations• Basic transformations:
– Translation– Scaling– Rotation
• Purposes:– To move the position of objects– To alter the shape / size of objects– To change the orientation of objects
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Basic two-dimensional geometric transformations (1/1)
• Two-Dimensional translation– One of rigid-body transformation, which move objects without
deformation– Translate an object by Adding offsets to coordinates to generate
new coordinates positions– Set tx,ty be the translation distance, we have
– In matrix format, where T is the translation vector
xtx'x yty'y
yx
P
y
x
tt
T
'y'x
'P
TP'P
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
– We could translate an object by applying the equation to every point of an object.• Because each line in an object is made up of an
infinite set of points, however, this process would take an infinitely long time.
• Fortunately we can translate all the points on a line by translating only the line’s endpoints and drawing a new line between the endpoints.
• This figure translates the “house” by (3, -4)
Basic two-dimensional geometric transformations (1/2)
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Translation Example
y
x0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
(1, 1) (3, 1)
(2, 3)
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Basic two-dimensional geometric transformations (2/1)
• Two-Dimensional rotation– Rotation axis and angle are specified for rotation– Convert coordinates into polar form for calculation
– Example, to rotation an object with angle a• The new position coordinates
• In matrix format
• Rotation about a point (xr, yr)
cosrx sinry
cossinsincos
R PR'P
cossincossinsincos)sin('sincossinsincoscos)cos('
yxrrryyxrrrx
cos)(sin)('sin)(cos)('
rrr
rrr
yyxxyyyyxxxx
rr
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
– This figure shows the rotation of the house by 45 degrees.
• Positive angles are measured counterclockwise (from x towards y)
• For negative angles, you can use the identities:– cos(-) = cos() and sin(-)=-sin()
Basic two-dimensional geometric transformations (2/2)
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Rotation Example
y
0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
(3, 1) (5, 1)
(4, 3)
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Basic two-dimensional geometric transformations (3/1)
• Two-Dimensional scaling– To alter the size of an object by multiplying the coordinates
with scaling factor sx and sy
– In matrix format, where S is a 2by2 scaling matrix
– Choosing a fix point (xf, yf) as its centroid to perform scaling
xsx'x ysyy
yx
s00s
'y'x
y
x PS'P
)s1(ysy'y)s1(xsx'x
yfy
xfx
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
– In this figure, the house is scaled by 1/2 in x and 1/4 in y• Notice that the scaling is about the origin:
– The house is smaller and closer to the origin
Basic two-dimensional geometric transformations (3/2)
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Scaling
Note: House shifts position relative to origin
y
x 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
12
13
36
39
– If the scale factor had been greater than 1, it would be larger and farther away.
WATCH OUT: Objects grow and move!
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Scaling Example
y
0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
(1, 1) (3, 1)
(2, 3)
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Homogeneous Coordinates
• A point (x, y) can be re-written in homogeneous coordinates as (xh, yh, h)• The homogeneous parameter h is a non-zero value such that:
• We can then write any point (x, y) as (hx, hy, h)• We can conveniently choose h = 1 so that (x, y) becomes (x, y, 1)
hxx h h
yy h
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Why Homogeneous Coordinates?
• Mathematicians commonly use homogeneous coordinates as they allow scaling factors to be removed from equations • We will see in a moment that all of the transformations we discussed previously can be represented as 3*3 matrices• Using homogeneous coordinates allows us use matrix multiplication to calculate transformations – extremely efficient!
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Homogenous Coordinates
• Combine the geometric transformation into a single matrix with 3by3 matrices
• Expand each 2D coordinate to 3D coordinate with homogenous parameter
• Two-Dimensional translation matrix
• Two-Dimensional rotation matrix
• Two-Dimensional scaling matrix
1yx
100t10t01
1'y'x
y
x
1yx
1000cossin0coscos
1'y'x
1yx
1000s000s
1'y'x
y
x
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Inverse transformations
• Inverse translation matrix
• Two-Dimensional translation matrix
• Two-Dimensional translation matrix
100t10t01
T y
x1
1000cossin0sincos
R 1
100
0s10
00s1
Sx
x
1
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (1)
• Composite transformation– A sequence of transformations– Calculate composite transformation matrix rather than
applying individual transformations
• Composite two-dimensional translations– Apply two successive translations, T1 and T2
– Composite transformation matrix in coordinate form
PM'PPMM'P 12
PTTPTTP
ttTT
ttTT
yx
yx
)()('
),(
),(
1212
222
111
),(),(),( 21211122 yyxxyxyx ttttTttTttT
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (2)
• Composite two-dimensional rotations– Two successive rotations, R1 and R2 into a point P
– Multiply two rotation matrices to get composite transformation matrix
• Composite two-dimensional scaling
PRRPPRRP
)}()({'
})({)('
12
12
PssssSP
ssssSssSssS
yyxx
yyxxyxyx
),('
),(),(),(
2121
21212211
PRPRRR
)('
)()()(
21
2112
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (3)
• General two-dimensional Pivot-point rotation– Graphics package provide only origin rotation– Perform a translate-rotate-translate sequence
• Translate the object to move pivot-point position to origin
• Rotate the object• Translate the object back to the original position
– Composite matrix in coordinates form),y,x(R)y,x(T)(R)y,x(T rrrrrr
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (4)
• Example of pivot-point rotation
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Pivot-Point Rotation
100sin)cos1(cossinsin)cos1(sincos
1001001
1000cossin0sincos
1001001
rr
rr
r
r
r
r
xyyx
yx
yx
,,,, rrrrrr yxRyxTRyxT
Translate Rotate Translate
(xr,yr)
(xr,yr)
(xr,yr)
(xr,yr)
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (5)
• General two-dimensional Fixed-point scaling– Perform a translate-scaling-translate sequence
• Translate the object to move fixed-point position to origin
• Scale the object wrt. the coordinate origin• Use inverse of translation in step 1 to return the object
back to the original position– Composite matrix in coordinates form
)s,s,y,x(S)y,x(T)s,s(S)y,x(T yxffffyxff
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (6)
• Example of fixed-point scaling
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
General Fixed-Point Scaling
100)1(0)1(0
1001001
1000000
1001001
yfy
xfx
f
fx
f
f
syssxs
yx
ss
yx
y
Translate Scale Translate
(xr,yr)
(xr,yr)
(xr,yr)
(xr,yr)
yxffffyxff ssyxSyxTssSyxT ,,,, ,,
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Two-dimensional composite transformation (7)
• General two-dimensional scaling directions– Perform a rotate-scaling-rotate sequence– Composite matrix in coordinates form
1000cosssinssincos)ss(0sincos)ss(sinscoss
)(R)s,s(S)(R
22
2112
122
22
1
211
s1
s2
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
General Scaling Directions
• Converted to a parallelogram
1000cossinsincos)(0sincos)(sincos
)(),()( 22
2112
122
22
1
211
ssss
ssssRssSR
x
y
(0,0)
(3/2,1/2)
(1/2,3/2)
(2,2)
x
y
(0,0)
(1,0)
(0,1)
(1,1)
x
y s2
s1
Scale
Computer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved © 2005 Pearson Education
Basic transformations such as translation, rotation and scaling are include in most graphics package. Some package provide additional transformations that are useful in certain applications. Two such transformations are reflection and shear.
A reflection is a transformation that produces a mirror image of an object.
The mirror image generated by rotating an object 180 deg about reflection axis
We can choose axis of reflection in the xy plane or perpendicular to xy plane.
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