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Graphics Programming: Transformations

This documentation explores fundamental graphics transformations such as translations, rotations, scaling, and reflections (mirror transformations) in both 2D and 3D spaces. It discusses how these transformations are utilized for scene composition, creating symmetrical objects, and enabling complex animations by moving objects relative to one another. An insight into matrix representations is provided, including mapping points through transformation matrices. A specific example is illustrated through window to viewport transformation, highlighting the practical applications of these concepts in graphics programming.

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Graphics Programming: Transformations

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  1. 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

  13. Combining Transforms

  14. 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 Q P

  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 2 3 4 4 7 5 5 2 8 1 0 0 0 1 0 0 0 1 2 3 4 4 7 5 5 2 8 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

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