1. April 20101 Geometric Transformations for Computer Graphics Shmuel Wimer Bar Ilan Univ., School of Engineering

2. April 20102 2D Translation

3. April 20103 2D Rotation

4. April 20104 2D Scaling

5. April 20105 Homogeneous Coordinates

6. April 20106 2D Translation 2D Rotation 2D Scaling

7. April 20107 Inverse transformations: Composite transformations: Composite translations:

8. April 20108 Composite Rotations: Composite Scaling:

9. April 20109 General 2D Rotation Move to originRotate Move back

10. April 201010 General 2D Scaling Move to originScale Move back

11. April 201011 2D Directional Scaling

12. April 201012 2D Reflections

13. April 201013 3 1 2 1 2 3 1 2 3 1 2 3

14. April 201014 Geometric Transformations by Rasterization The transformed shape needs to be filled. –A whole scan-line filling is usually in order. However, simple transformations can save new filling by manipulating blocks in the frame buffer. Translation: Move block of pixels of frame buffer into new destination.

15. April 201015 90° counterclockwise rotation 180° rotation Rotated pixel block Destination pixel array RGB of destination pixel can be determined by averaging rotated ones (as antialiasing)

16. April 201016 Translation 3D Transformations Very similar to 2D. Using 4x4 matrices rather than 3x3.

17. April 201017 General 3D Rotation 1.Translate the object such that rotation axis passes through the origin. 2.Rotate the object such that rotation axis coincides with one of Cartesian axes. 3.Perform specified rotation about the Cartesian axis. 4.Apply inverse rotation to return rotation axis to original direction. 5.Apply inverse translation to return rotation axis to original position.

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19. April 201019

20. April 201020

21. April 201021

22. April 201022

23. April 201023 Efficient 3D Rotations by Quaternions

24. April 201024

25. April 201025 3D Scaling Enlarging object also moves it from origin

26. April 201026 Scaling with respect to a fixed point (not necessarily of object)