3d object representation -...

Graphics

Graphics Lab @ Korea University

Geometric

Transformation

Korea University

http://kucg.korea.ac.kr

KUCG


Transformation

Model Transformation

Lighting

Viewing Transformation

Projection Transformation

Clipping

Viewport Transformation

Scan Conversion

Image

3D Primitives

Transform into 3D World Coordinate System

Illustrate According toLighting and Reflectance

Transform into 3D Viewing Coordinate System

Transform into 2D Viewing Coordinate System

Clip Primitives outside Window’s View

Transform into Viewport

Draw Pixels


KUCG


Transformation

3D Object Coordinate

Model Transformation

Viewing Transformation

Projection Transformation

Viewport Transformation

p(x’, y’) or p(u, v)

P(x, y, z)

3D World Coordinate

3D Viewing Coordinate

2D Projection Coordinate

2D Device Coordinate

3D Viewing Coordinate

3D Object Coordinate

3D World Coordinate


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Contents

Translation

Scaling

Rotation

Other Transformations

Coordinate Transformations

OpenGL Transformations


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Transformation in 3D

Transformation matrix

33 : Scaling, Reflection, Shearing, Rotation

31 : Translation

14 : Homogeneous Representation

1131

1333

000 S

LIFC

KHEB

JGDA

1×4


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3D Translation

zyx tzztyytxx ',','

11000

100

010

001

1

'

'

'

z

y

x

t

t

t

z

y

x

z

y

x

xz

y


KUCG


3D Scaling

zyx szzsyysxx ',','

11000

000

000

000

1

'

'

'

z

y

x

s

s

s

z

y

x

z

y

x

xz

y


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 x x xzzzz

y y y y

Original Position Translation Scaling Inverse Translation


KUCG


3D Rotation

Coordinate-axes rotations

x-axis rotation

y-axis rotation

z-axis rotation

General 3D rotations

Rotation about an axis that is parallel to one of the

coordinate axes

Rotation about an arbitrary axis


KUCG


Coordinate-axes Rotations

z-Axis Rotation x-Axis Rotation y-Axis Rotation

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


KUCG


Order of Rotations

Order of rotations affects final position


KUCG


General 3D Rotations

Rotation about an axis that is parallel to one of the coordinate axes


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Rotation about an arbitrary axis

Basic Idea

1. Translate (x1, y1, z1) to the origin

2. Rotate (x’2, y’2, z’2) on to the z axis

3. Rotate the object around the z-axis

4. Rotate the axis to the original

orientation

5. Translate the rotation axis to the

original position

(x2, y2, z2)

(x1, y1, z1)

x

y

R-1

T-1

R

T

TRRRRRTR xyzyx

111

R

z


KUCG



Step 1. Translate

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)

ProjectedPoint

RotatedPoint

x

y

z


KUCG


Arbitrary Axis Rotation

Step 3. Rotate about y axis by β (clockwise)

222222

cos,sin

dacbal

l

d

l

a

1000

0/0/

0010

0/0/

1000

0cos0sin

0010

0sin0cos

ldla

lald

y

R

x

y

z

ProjectedPoint

RotatedPoint

d

(a,0,d)


KUCG



Step 4. Rotate about z axis by the angle

1000

0100

00cossin

00sincos

zR

y

x

z

l


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Step 5. Reverse transformation

To place the axis back in its initial position

x

y

z

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

1000

100

010

001

1

1

1

111

z

y

x

yx RRT

TRRRRRTR xyzyx

111

l

l


KUCG


Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2, 1, 0) and B(3, 3, 1).

Example

A Unit Cube


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Example

Step1. Translate point A to the origin

T(-2, -1, 0)

1000

0100

1010

2001

T

x

z

y

A’(0, 0, 0)

B’(1, 2, 1)


KUCG


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

Example

Step 2. Rotate axis A’B’ about the x axis by

Until it lies on the xz plane

x

z

y

l

(0, 2, 1)

B”(1, 0, )

B’(1, 2, 1)

5


KUCG


x

z

y

1000

06

300

6

6

0010

06

60

6

30

yR

6

30

6

5cos,

6

6

6

1sin

Example

Step 3. Rotate axis A’B’’ about the y axis by β

Until it coincides with the z axis

(0, 0, 6)

l

B”(1, 0, )5


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Example

Step 4. Rotate the cube 90° about the z axis

Finally, the concatenated rotation matrix about the

arbitrary axis AB becomes,

TRRRRRTR xyzyx 90111

1000

0100

0001

0010

90zR


KUCG


1000

560.0167.0741.0650.0

151.1075.0667.0742.0

742.1983.0075.0166.0

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

Example


KUCG


PRP

Example

Multiplying R(θ) by the point matrix of the original cube

11111111

076.0091.0560.0726.0817.0650.0301.1467.1

483.0409.0151.1225.1184.0258.0484.0558.0

891.2909.1742.1725.2816.2834.1667.1650.2

11111111

10011001

00001111

11001100

1000

560.0167.0741.0650.0

151.1075.0667.0742.0

742.1983.0075.0166.0

P


KUCG


11000

0100

010

001

1

'

'

'

z

y

x

b

a

z

y

x

11000

0100

0010

0001

1

'

'

'

z

y

x

z

y

x

Other Transformations

Reflection to the xy plane

z-axis shear

xz

y

xz

y

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


KUCG



Coordinate systems

Local (modeling) coordinate

World coordinate

Scene

<Local Coordinate System>


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Coordinate systems

Local (modeling) coordinate

World coordinate

Scene

<Word Coordinate System>


KUCG



void glRotatef(GLfloat angle, GLfloat x, GLfloat y, GLfloat z)

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glRotatef(45.0, 0.0, 0.0, 1.0);

glutSolidCube(0.3);


KUCG



void glTranslatef(GLfloat dx, GLfloat dy, GLfloat dz)


glLoadIdentity();

glTranslatef(0.5, -0.5, 0.0);

glutSolidCube(0.3);


KUCG



void glScalef(GLfloat sx, GLfloat sy, GLfloat sz)


glLoadIdentity();

glScalef(2.0, 1.0, 1.0);

glutSolidCube(0.3);


KUCG



OpenGL Transformation Sample


3d object representation -...

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