Geometric Transformationskucg.korea.ac.kr/.../tutor/04GeometricTransformation.pdfKUCG Graphics Lab @ Korea University Homogeneous Coordinate in Graphics Translation matrix is integrated

Graphics

Graphics Lab @ Korea University http://kucg.korea.ac.kr

Geometric

Transformations

Korea Univ.

Computer Graphics Lab.

KUCG

Graphics Lab @ Korea University

Geometric Transformations

Translation

Scaling

Rotation

etc.

http://kucg.korea.ac.kr

KUCG


Geometric Transformations’ Role

Positioning

Deformation

Animation

Camera Control

etc.


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3D Translation

11000

100

010

001

1

'

'

'

z

y

x

t

t

t

z

y

x

z

y

x

x

z

y

zyx tzztyytxx ' ,' ,'

T(tx, ty, tz)


KUCG


3D Scaling

11000

000

000

000

1

'

'

'

z

y

x

s

s

s

z

y

x

z

y

x

x

z

y

zyx szzsyysxx ' ,' ,'


KUCG


Fixed-point Scaling

11000

100

010

001

1000

000

000

000

1000

100

010

001

1

'

'

'

),,( ),,( ),,(z

y

x

z

y

x

s

s

s

z

y

x

z

y

x

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x z

y

Original Position Translation Scaling Inverse Translation

x z

y

x z

y

x z

y

F


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3D Rotations

Rotations about Basic Coordinate Axes

x-axis rotation

y-axis rotation

z-axis rotation

General 3D Rotations

About an axis parallel to one of basic coordinate axes

About an arbitrary axis


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Order of Rotations


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Rotations about Basic Coordinate Axes

11000

0100

00cossin

00sincos

1

'

'

'

z

y

x

z

y

x

11000

0cossin0

0sincos0

0001

1

'

'

'

z

y

x

z

y

x

11000

0cos0sin

0010

0sin0cos

1

'

'

'

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z-axis Rotation x-axis Rotation y-axis Rotation


KUCG


Homogeneous Coordinate

4th coordinate in 3D means

In Mathematics

The size of basic step of the coordinate system

ex) (1, 2, 3, 1) = (2, 4, 6, 2)

In Graphics

Translation matrix into the multiplication scheme


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Homogeneous Coordinate in

Graphics

Translation matrix is integrated into the

multiplication scheme

Provides computational efficiency

ex) a complex object with millions of vertices

To be scaled, rotated, and translated

Two possible choices

Applying 3 transformation matrices to every vertex

Applying just a pre-computed composite matrix

Careful with the order,


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General 3D Rotation Type 1

About an axis parallel to one of basic coordinate axes


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General 3D Rotation Type 2

About an Arbitrary Axis

(x2, y2, z2)

(x1, y1, z1)

x

y

RA-1

T-1

RA

T

R

z

Translate (x1, y1, z1) to the origin

Rotate (x2’, y2’, z2’) on to the z axis

Rotate the object around the z-axis

Rotate the axis to the original orientation

Translate the rotation axis to the original position


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Arbitrary Axis Rotation Process(1/5)

Step 1. Translate the Tail Vertex to the Origin

1000

100

010

001

1

1

1

z

y

x

T

x

y

z

(x2, y2, z2)

(x1, y1, z1)


KUCG



Step 2. Rotate about x Axis by α

1000

0//0

0//0

0001

1000

0cossin0

0sincos0

0001

dcdb

dbdcx

R

d

c

cb

c

d

b

cb

b

22

22

cos

sin(a, b, c)

(0, b, c)

Projected

Point

Rotated

Point

x

y

z

121212 , , zzcyybxxa


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Step 3. Rotate about y axis clockwisely by β

Counterclockwisely by -β

l

d

l

a

dacbal

cos ,sin

222222

1000

0/0/

0010

0/0/

1000

0cos0sin

0010

0sin0cos

1000

0cos0sin

0010

0sin0cos

ldla

lald

y

R

Projected

Point

Rotated

Point

d

(a,0,d)

x

y

z


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Step 4. Rotate about z Axis by

1000

0100

00cossin

00sincos

zRx

y

z


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Step 5. Reverse Transformation

To place the axis back in its initial position

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

1000

100

010

001

1

1

1

111

z

y

x

yx RRT

x

y

z

l

l

TRRRRRTR )()()()()()( 111 xyzyx


KUCG


Characteristics of the Rotation Matrix

Orthogonal

Each row vectors are perpendicular

Each column vectors are perpendicular

Orthonormal = Orthogonal + …

Each row vector is a unit vector

Each column vector is a unit vector

All rotation matrices should be “orthonormal”

Inverse of Orthonormal Matrix

The transpose of the original matrix

TRR 1


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Other Transformations

Reflection to the xy Plane

z-axis Shear

x z

y

11000

0100

0010

0001

1

'

'

'

z

y

x

z

y

x

x z

y

(a, b, c) (a, b, -c)

11000

0100

010

001

1

'

'

'

z

y

x

b

a

z

y

x


KUCG


3D Transformation Summary

General Transformation Matrix

4x4 square matrix

1141

1333

000 S

LIFC

KHEB

JGDA Scaling, Reflection,

Shearing, Rotation Translation


Graphics


Arbitrary Axis Rotation

Example


KUCG


Goal

Find the rotation matrix that rotates 90º about

an axis by its endpoints A(2, 1, 0) and B(3, 3, 1)

Works to Do

Translate the point A to the origin

T(-2, -1, 0)

Adjust the axis AB to one of the coordinate axes

Rotate 90º

Reverse all the transformations


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Step 1

Translate tail point A to the origin

T(-2, -1, 0)

1000

0100

1010

2001

Tx

z

y

A’(0, 0, 0)

B’(1, 2, 1)


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Step 2

Rotate axis A’B’ about the x axis by

Until it lies on the xz plane

1000

05

5

5

520

05

52

5

50

0001

xR

6121,5

5

5

1cos,

5

52

5

2

12

2sin 222

22

l

x

z

y

l

(0, 2, 1)

B”(1, 0, )

B’(1, 2, 1)

5


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Step 3

Rotate axis A’B’’ about the y axis by β

Until it coincides with the z axis

x

z

y

1000

06

300

6

6

0010

06

60

6

30

yR

6

30

6

5cos,

6

6

6

1sin

(0, 0, )

l

B”(1, 0, ) 56


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Step 4

Rotate 90° about the z axis

1000

0100

0001

0010

90zz RR

TRRRRRTR xyzyx 111


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Final Result

1000

558.0167.0741.0650.0

149.1075.0667.0741.0

742.1983.0075.0167.0

1000

0100

1010

2001

1000

05

5

5

520

05

52

5

50

0001

1000

06

300

6

6

0010

06

60

6

30

1000

0100

0001

0010

1000

06

300

6

6

0010

06

60

6

30

1000

05

5

5

520

05

52

5

50

0001

1000

0100

1010

2001

R

477.530

449.26

236.25


Geometric Transformationskucg.korea.ac.kr/.../tutor/04GeometricTransformation.pdfKUCG Graphics Lab @ Korea University Homogeneous Coordinate in Graphics Translation matrix is integrated

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