1. 3D Geometry and Transformations11
고려대학교 컴퓨터학과
김 창 헌

2. Contents Translation Scaling Rotation Other transformations
Transformation of coordinate systems

3. Transformation in 3D 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 x z y
Original position
Translate
Scaling
Inverse Translate

7. 3D Rotation 1. Coordinate-Axes Rotations
좌표축을 기준으로 회전
2. General Three-Dimensional Rotations
좌표축에 평행한 회전축 기준 회전
임의의 회전축(직선) 기준 회전

8. Coordinate-Axis Rotationsy
X축 중심 회전
Z 축 중심 회전 x z
x축 중심 회전 y z y x
z축 중심 회전
Y축 중심 회전 x z
y축 중심 회전

9. Order of rotations affects the final position of an object

10. Rotation about an Principal axis
 좌표축과 평행한 회전축 중심 회전
물체를 좌표축과 평행하게 이동 (회전축이동) 회전
물체를 원위치로 이동 (회전축 원위치)

11. Rotation about an arbitrary axis
 Basic Idea
1. 원점을 지나도록 회전축을
평행이동
2. 좌표축과 일치하도록
회전축을 회전
3. 축에 대한 회전
4. 회전축을 원래 방향으로
역회전
5. 회전축을 원위치로 평행 이동 y T
(x2,y2,z2) R
(x1,y1,z1)
R-1 x
T-1 z

12. Rotation about an arbitrary axis
Step 1. Translation
(x2,y2,z2)
(x1,y1,z1) x z y

13. Rotation about an arbitrary axis
Step 2. Establish [ TR ]x x axis y
(0,b,c)
(a,b,c)
Projected Point   x z
Rotated Point

14. Rotation about an arbitrary axis
Step 3. Rotate about y axis by  y
(a,b,c) l
Projected Point d x 
(a,0,d)
Rotated Point z

15. Rotation about an arbitrary axis
Step 4. Rotate about z axis by the desired angle  y l x  z

16. Rotation about an arbitrary axis
Step 5. Apply the reverse translation to place the
axis back in its initial position x z y l

17. Rotation about an arbitrary axis
Ex) Find the new coordinates of a unit cube rotated about
an axis defined by its endpoints A(2,1,0) and B(3,3,1).
Step1. Translate point A to the origin y
B’(1,2,1)
A’(0,0,0) x
A Unit Cube z

18. Rotation about an arbitrary axis
Step 2. Rotate axis A’B’ about the x axis by and 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)

19. Rotation about an arbitrary axis
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,  6) z

20. Rotation about an arbitrary axis
Step 4. Rotate the cube 90° about the z axis
Finally, the concatenated rotation matrix about the arbitrary
axis AB becomes,

21. Rotation about an arbitrary axis

22. Rotation about an arbitrary axis
Multiplying [TR]AB by the point matrix of the original cube

23. 3D Reflections & Shears Reflection relative to the xy plane
z-axis shear y y z z x x

24. Transformation of Coordinate System
Front-Wheel Tractor System World
Coordinate Coordinate Coordinate

25. Transformation of Coordinate System
Use of Multiple Coordinate System
zworld