Lecture 8

3D Geometric Transformation

Applying transformation matrices

D

nnn

D

nn

zyx

zyx

zyx

zyx

zyx

zyx

yx

yx

yx

yx

yx

yx

nnn

nn

3

222

111

'''

'''

'''

2

22

11

''

''

''

1

1

1

1

1

1

1

1

1

1

1

1

222

111

22

11

Translation

1

010

001

yx

TR

1

0100

0010

0001

zyx

TR

2D Translation

3D Translation

•Component-wise addition of vectors

x’ = x + dx y’ = y + dyz’ = z + dz

•To move polygons: translate vertices (vectors) and redraw lines between them•Preserves lengths and angles•Does not require reference point

Rotation

100

0cossin

0sincos

RRotation 2D

•Rotation of vectors through an angle

x’ = x cos Ө – y sin Өy’ = x sin Ө + y cos Ө

•Preserves lengths and angles•Point of rotation point (0,0)

(X,Y)

A

(X cos - Y sin , x sin + Y cos)A’

Center of

rotation

(0,0)

Rotation at other point than origin point

Let say that the center of rotation is (x1, y1). Translation has to be carried before and after rotation

Procedure1. Convert the (x1, y1) to (0,0) by applying translation x

= -x1 and y= -y1. 2. Apply the rotation matrix3. Convert the (0,0) to its original point (x1, y1) by

applying translation x = x1 and y = y1.

111

010

001

111

010

001

1

1

1

1

1

1

22

11

''

''

''

22

11

yxyxyx

yx

yx

yx

yx

yx

R

nnnn

3D Rotation

1000

0cossin0

0sincos0

0001

XR

1000

0cos0sin

0010

0sin0cos

YR

1000

0100

00cossin

00sincos

ZR

Axis of rotation: x

Axis of rotation: z

Axis of rotation: y

If axis of rotation other than specified axis: apply translation before and after 3D rotation is applied

Foundation of transformations

Similar method can be applied on other transformation such as shear, reflection etc.

Important, if the transformation requires point (2D), axis (2D & 3D) and plane (3D) at the origin CS. Translation must be applied before and after applying the respective transformation.

Applying transformation composite matrices

Do this transformation on screen1. Draw rectangle any size

2. Translate dx = 6

3. Rotate it by 450

1. The same rectangle

2. Rotate it 450

3. Translate dx = 6

Q: Do both rectangles overlapping each other? Why?

Y

X0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

Y

X0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

Translation → Rotation

Rotation → Translation

Composite Transformations

Composite Transformation

C

nC

XX

'

321 ...

Composite Transformation:

Combination of multiple transformations must be in sequence.

Example 1

A cube (1 x 1 x 1 ) as shown in Figure 2a is transformed into the solid as shown in Figure 2b. Establish the transformation matrices for this transformation.

The scaling matrix transformation based on reference point of (0, 0, 0) is as in the following.

1000

000

000

000

Z

y

x

SS

S

S

Example 1: Solution

1004

0100

0010

0001

1000

0100

0040

0001

[Scaling][Translation]

Rename 3Dsolidcube.cpp and try this problem• Note: 1. Matrix in openGL is applied in reverse sequence, last will be applied first and first will be applied last.2. By assign variable to the scale matrix and this variable will be controlled by the idle function, show an animation of the cube becomes bigger.

Example 2

The available API function allows only to draw cube with ( l x l x l ) size.Based on this function and the transformation matrices, model the robotic arm with arm of ( 2 x 2 x 20) and (2 x 2 x 5) at the position as shown the figure. The center bottom of the lower arm is at ( 0, 0, 0).

Discuss the methodology to model both lower and upper arm to be positioned as in the figure

Step 1: Scaling to lower and upper

1000

02000

0020

0002

*1

armLLarm

The scaling of lower arm

The scaling of the

upper arm

1000

0500

0020

0002

*1

armUU arm

Example 2

Step 2: Translate the upper arm at the end of the lower arm

12000

0100

0010

0001

*1*2

armUU arm

Step 3: Rotate lower and upper arm along +ve Y axis by 30o

1000

0)30cos(0)30sin(

0010

0)60sin(0)30cos(

*1*3

oo

oo

armarm LL

The transformation for the

lower arm.

The transformation for the upper arm

1)60sin(200)60cos(20

0100

0010

0001

1000

0)30cos(0)30sin(

0010

0)30sin(0)30cos(

1)60sin(200)60cos(20

0100

0010

0001

*2*3

oo

oo

oo

oo

armarm UU

Example 2

Step 4: Rotate upper arm along +ve Y axisby 10o

1)60sin(200)60cos(20

0100

0010

0001

1000

0)10cos(0)10sin(

0010

0)10sin(0)10cos(

1)60sin(200)60cos(20

0100

0010

0001

*3*4

oo

oo

oo

oo

armarm UU

The center rotation is at the end of the lower

arm which is (20 cos(60o), 20 sin(60o)).

Therefore, translation must be applied

before and after the rotation matrix.

Example 2

Step 5: Rotate the lower and upperarm by 15o along +ve Z axis

1000

0100

00)15cos()15sin(

00)15sin()15cos(

*3*5

oo

oo

armarm LL

The transformation for the

lower arm.

The transformation for the

upper arm.

1)60sin(200)60cos(20

0100

0010

0001

1000

0100

00)15cos()15sin(

00)15sin()15cos(

1)60sin(200)60cos(20

0100

0010

0001

*4*5

oo

oo

oo

oo

armarm UU

Example 2

Example 3: Planet and Sun

You have to do animation of planet goes around sun using two solid sphere.

Rename 3Dcubesolid.cpp and modify the program to model the sun and the planet using sphere with 5 and 2 unit diameter respectively. The planet orbit is 10 unit radius.