1. Geometrical Transformations
Prof. Lizhuang Ma
Shanghai Jiao Tong University

2. Contents 2D Transformations Homogeneous Coordinates &
Matrix Representation
The Window-to-Viewport Transformation
3D Transformations

3. 2D Transformations
2D Translation
2D Scaling
2D Rotation

4. 2D Translation

5. 2D Scaling

6. 2D Rotation

7. Homogeneous Coordinates
Why & What is homogeneous coordinates?
if points are expressed in homogeneous coordinates, all three transformations can be treated as multiplications:

8. Homogeneous Coordinates

9. Homogeneous Visualization
Divide by w to normalize (homogenize)
W = 0? Point at infinity (direction)
w= 1, w= 2
(0, 0, 1) = (0, 0, 2) = …
(7, 1, 1) = (14, 2, 2) = …
(4, 5, 1) = (8, 10, 2) = …

10. Homogeneous Coordinates 2D Translation

11. Homogeneous Coordinates for 2D Translation

12. Homogeneous Coordinates 2D Scaling

13. Homogeneous Coordinates 2D Rotation

14. Properties of Transformations
Rigid-body transformations
Rotation & translation
Preserving angles and lengths
Affine transformations
Rotation & translation & scaling
Preserving parallelism of lines
Shear transformation

15. 2D Shear Transformation

16. Composition of 2D Transformations

17. The Window-to-Viewport Transformation

18. The Window-to-Viewport Transformation

19. 3D Translation & 3D Scaling

20. 3D Rotations

21. 3D Shear