3-D Geometric Transformations

3-D GeometricTransformations

Some of the material in these slides may have been adapted from the lecture notes of Graphics Lab @ Korea University

2

Contents

● Translation● Scaling ● Rotation● Other Transformations● Coordinate Transformations

3

Transformation in 3D

● Transformation Matrix

1131

1333

000 SLIFCKHEBJGDA

33 : Scaling, Reflection, Shearing, Rotation31 : Translation

13 : Homogeneous representation11 : Uniform global Scaling

4

3D Translation

● Translation of a Point

zyx tzztyytxx ',','

11000100010001

1'''

zyx

ttt

zyx

z

y

x

xz

y

5

3D Scaling

● Uniform Scaling

zyx szzsyysxx ',','

11000000000000

1'''

zyx

ss

s

zyx

z

y

x

xz

y

6

Relative Scaling

● Scaling with a Selected Fixed Position

11000100010001

1000000000000

1000100010001

1'''

),,(),,(),,(zyx

zyx

ss

s

zyx

zyx

zyxTsssSzyxTf

f

f

z

y

x

f

f

f

fffzyxfff

x x x xzzzz

y y y y

Original position Translate Scaling Inverse Translate

7

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

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Rotation is counterclockwise

9

Coordinate-Axes Rotations

Z-Axis Rotation ● X-Axis Rotation ● Y-Axis Rotation

11000010000cossin00sincos

1'''

zyx

zyx

z

y

x

110000cossin00sincos00001

1'''

zyx

zyx

110000cos0sin00100sin0cos

1'''

zyx

zyx

z

y

xz

y

x

10

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3D line equation

● Given A (x0, y0, z0) and B (x1, y1, z1) forming a line.● The equation of this line can be formulated as follows:

ddn

zzyyxx

dcba

n

cba

tzyx

zyx

ˆ,010101

,

.000

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Order of Rotations

● Order of Rotation Affects Final Position■ X-axis Z-axis

■ Z-axis X-axis

13

General 3D Rotations

● Rotation about an Axis that is Parallel to One of the Coordinate Axes■ Translate the object so that the rotation axis coincides

with the parallel coordinate axis■ Perform the specified rotation about that axis■ Translate the object so that the rotation axis is moved back

to its original position

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Rotation about an arbitrary axis

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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

axis3. 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

z

y

R-1

T-1

R

T

TRRRRRTR xyzyx111

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● Step 1. Translation

1000100010001

1

1

1

zyx

T

(x2,y2,z2)

(x1,y1,z1)

x

z

y

17


● Step 2. Establish [ TR ]x x

axis

10000//00//00001


dcdbdbdc

x

R

(a,b,c)(0,b,c)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z22 cb

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● Step 2. Establish [ TR ]x x

axis

22 cb

(a,b,c)(0,b,c)

Projected Point

Rotated Point

dc

cb

cdb

cb

b

22

22

cos

sin

x

y

z

b

c

22 cb

a

222 cba

(a,b,c)(0,b,c)

(0,0,0)(0,0,0)

(0,b,c)

(0,0,c)

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Arbitrary Axis Rotation

● Step 3. Rotate about y axis by

(a,b,c)

(a,0,d)

l

d

22

222222

cos,sin

cbd

dacballd

la

10000/0/00100/0/


ldla

lald

y

Rx

y

Projected Point

zRotated

Point

a

(0,0,0)

(a,0,d)

d

(0,0,d)

l

20


● Step 4. Rotate about z axis by the desired angle

l

1000010000cossin00sincos

zR

y

x

z

21


● Step 5. Apply the reverse transformation to place the axis back in its initial position

x

y

l

l

z



1000100010001

1

1

1

111

zyx

yx RRT

TRRRRRTR xyzyx111

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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).

A Unit Cube

Example

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Example

● Step1. Translate point A to the origin

A’(0,0,0)x

z

y

B’(1,2,1)

1000010010102001

T

24

xz

y

l

1000

055

5520

05

52550

0001

xR

6121

55

51cos

552

52

12

2sin

222

22

lB’(1,2,1)

Projected point (0,2,1)

B”(1,0,5)

Example

● Step 2. Rotate axis A’B’ about the x axis by an angle , until it lies on the xz plane.

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x

z

y

l

1000

06300

66

0010

0660

630

yR

630

65cos

66

61sin

B”(1,0, 5)(0,0,6)

Example

● Step 3. Rotate axis A’B’’ about the y axis by and angle , until it coincides with the z axis.

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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

1000010000010010

90zR

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1000560.0167.0741.0650.0151.1075.0667.0742.0

742.1983.0075.0166.0

1000010010102001

1000

055

5520

05

52550

0001

1000

06300

66

0010

0660

630

1000010000010010

1000

06300

66

0010

0660

630

1000

055

5520

05

52550

0001

1000010010102001

R

Example

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PRP

11111111076.0091.0560.0726.0817.0650.0301.1467.1483.0409.0151.1225.1184.0258.0484.0558.0

891.2909.1742.1725.2816.2834.1667.1650.2

11111111100110010000111111001100

1000560.0167.0741.0650.0151.1075.0667.0742.0

742.1983.0075.0166.0

P

Example

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

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Other Transformations

● Reflection Relative to the xy Plane

● Z-axis Shear

xz

y

x

z

y

11000010000100001

1zyx

zyx

'''

110000100010001

1'''

zyx

ba

zyx

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Coordinate Transformations

● Multiple Coordinate System■ Local (modeling) coordinate system ■ World coordinate scene

Local Coordinate System

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● Multiple Coordinate System■ Local (modeling) coordinate system ■ World coordinate scene

Word Coordinate System

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● Example – Simulation of Tractor movement■ As tractor moves, tractor coordinate system and front-

wheel coordinate system move in world coordinate system

■ Front wheels rotate in wheel coordinate system

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● Transformation of an Object Description from One Coordinate System to Another

● Transformation Matrix■ Bring the two coordinates systems into alignment1. Translation

x

y

z (0,0,0)

y’

z’

x’u'y

u'z

u'x(x0,y0,z0)x

y

z (0,0,0)

u'y

u'z

u'x

000 , , zyx T

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2. Rotation & Scaling

x

y

z (0,0,0)

y’

z’

x’u'y

u'z

u'x x

y

z (0,0,0)

u'y

u'zu'x x

y

z (0,0,0)

u'y

u'zu'x

1000000

321

321

321

zzz

yyy

xxx

uuuuuuuuu

R

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3-D Geometric Transformations