1. ARCH 655 Parametric Modeling in Design Wei Yan, Ph.D., Associate Professor College of Architecture, Texas A&M University Lecture 5 Matrices and Transforms Acknowledgements: most materials in the slides are copies from Prof. Donald House’ The Digital Image, and some other materials are copies from various online resources.

2. Affine Maps or Warps General form of an affine map coefficients a ij are constants A geometric transformation that maps points and parallel lines to points and parallel lines

3. Affine Maps or Warps they can be represented in matrix form

4. Homogeneous Coordinate System (every point has an identical third coordinate)

5. Affine Maps (Transformations)

9. Recall TRIGONOMETRY Formula:

10. Composing Affine Warps Let R be a rotation, S be a scale, and T a translation. Let’s do a rotation, followed by a scale, followed by a translation.

11. Composing Affine Warps This is a wonderfully compact and unified way of constructing a large variety of useful warps in a very intuitive way – i.e. by simple composition of easy to understand operations. Matrix representation makes it simple for us to write and understand the transforms.

12. Example

13. Commutative?

15. Perspective Warps: Non-Affine Transformations Provides 3D feeling Giving depth illusion

16. Perspective Transformation Step1 - matrix multiplication, after which the third coordinate of the resulting point is not 1

17. Perspective Transformation Step 2, we need to restore our points to homogeneous coordinates with w = 1. We divide each vector by its own w coordinate

18. Example

23. If P is: 1 0 0 0 1 0 a b 1 Then the vanishing points will be (1/a, 0) and (0, 1/b). If a=0 and b=0 there will be no vanishing points (the image will be mapped to the same as the original). If one of them is 0, then there will be 1 vanishing point. Vanishing Points

24. Affine v.s. Perspective (Forward map) Affine: only one step

25. Affine v.s. Perspective (Forward map) Perspective: two steps

27. Projective Warps: Affine, Perspective or Composite of the two Affine and Perspective are unified here: both use the division by w. Affine becomes a special case, where w=1.