1. 3-D Geometric Transformations
Some of the material in these slides may have been adapted from the lecture notes of Graphics Korea University

2. Contents Translation Scaling Rotation Other Transformations
Coordinate Transformations

3. Transformation in 3D Transformation Matrix
33 : Scaling, Reflection, Shearing, Rotation
31 : Translation
11 : Uniform global Scaling
13 : Homogeneous representation

4. 3D Translation
Translation of a Point y x z

5. 3D Scaling
Uniform Scaling y x z

6. Relative Scaling Scaling with a Selected Fixed Position y y y y x x xz z z z
Original position
Translate
Scaling
Inverse Translate

7. 3D Rotation Coordinate-Axes Rotations General 3D 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

8. Rotation is counterclockwise

9. Coordinate-Axes Rotations
Z-Axis Rotation
X-Axis Rotation
Y-Axis Rotation y y y x x x z z z

11. 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:

12. 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

14. Rotation about an arbitrary axis

15. General 3D Rotations Rotation about an Arbitrary Axis Basic Idea T R
Translate (x1, y1, z1) to the origin
Rotate (x’2, y’2, z’2) 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 y T
(x2,y2,z2) R
(x1,y1,z1)
R-1 x z
T-1

16. General 3D Rotations
Step 1. Translation
(x2,y2,z2)
(x1,y1,z1) x z y

17. General 3D Rotations Step 2. Establish [ TR ]x x axis y (0,b,c)
(a,b,c)
Projected Point   x z
Rotated Point

18. General 3D Rotations Step 2. Establish [ TR ]x x axis b c y (0,b,c)
(a,b,c)
Projected Point   x
(0,b,c) a
(0,b,c)
(a,b,c) z
Rotated Point b 
(0,0,c)
(0,0,0) c
(0,0,0)

19. Arbitrary Axis Rotation
Step 3. Rotate about y axis by  y
(a,b,c)
Projected Point l d x 
(a,0,d)
Rotated Point z
(0,0,0)  l d
(a,0,d)
(0,0,d) a

20. Arbitrary Axis Rotation
Step 4. Rotate about z axis by the desired angle  y l x  z

21. Arbitrary Axis Rotation
Step 5. Apply the reverse transformation to place the axis back in its initial position y l l x z

22. Example
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

23. Example Step1. Translate point A to the origin y x z B’(1,2,1)

24. Example
Step 2. Rotate axis A’B’ about the x axis by an angle , until it lies on the xz plane. y
Projected point
(0,2,1)
B’(1,2,1) l  x z
B”(1,0,5)

25. Example
Step 3. Rotate axis A’B’’ about the y axis by and angle , until it coincides with the z axis. y l  x
(0,0,6)
B”(1,0,  5) z

26. Example Step 4. Rotate the cube 90° about the z axis
Finally, the concatenated rotation matrix about the arbitrary axis AB becomes,

27. Example

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

29. Other Transformations
Reflection Relative to the xy Plane
Z-axis Shear y y z z x x

30. Coordinate Transformations
Multiple Coordinate System
Local (modeling) coordinate system
World coordinate scene
Local Coordinate System

31. Coordinate Transformations
Multiple Coordinate System
Local (modeling) coordinate system
World coordinate scene
Word Coordinate System

32. Coordinate Transformations
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

33. Coordinate Transformations
Transformation of an Object Description from One Coordinate System to Another
Transformation Matrix
Bring the two coordinates systems into alignment
1. Translation y’ z’ x’
u'y
u'z
u'x
(x0,y0,z0) x y z
(0,0,0)
u'y x y z
(0,0,0)
u'x
u'z

34. Coordinate Transformations
2. Rotation & Scaling y’ x y z
(0,0,0) x y z
(0,0,0) x y z
(0,0,0) x’
u'y
u'y
u'y
u'x
u'x
u'x
u'z
u'z
u'z z’