1. Graphics Programming Graphics Programming: Transformations

2. Transformation (Isometry) Translations Rotations Scaling Mirror

3. Transformations One example of a transformation is the window to viewport transformation. Here we have seen an image in the world window scaled and translated (moved) into a viewport window. We can build on this transformation to allow us to move objects to more complex locations.

4. Transformations A Transformation consists of: a Rotation a Scaling and a Translation a Shearing They occur in 2D and 3D

5. Transformations Transformations allow for: scene composition

6. Transformations Transformations allow for: easily create symmetrical objects

7. Transformations Transformations allow for: viewing objects at different angles computer animation where several objects need to move relative to one another

8. Translation

9. Scaling

10. Rotation

11. Reflection (flip)

12. Combining Transforms

15. Transforming Points A transformation simply takes a point and maps it to another location.

16. Transforming Points In the 2D case this means…. Q = M(P) where M is some mapping matrix P Q

17. Matrices : Addition

18. Matrices : Multiplication 2x - y + 2z = 1 x + 2y - 4z = 3 3x - y + z = 0

19. Matrices : Multiplication

20. Matrices : Rotation

21. Matrices :Identity Matrix 100 0 10 0 01 234475528234475528 = 234475528234475528 x

22. Further Reading See Rowe – Chapter 2 for a discussion on homogenous coordinates and further examples of matrix transformations. Also see discussions and technical articles on: www.gamedeveloper.com www.gamedev.net www.ddj.com Dr. Dobb’s Journal - been around since 1976