SE 313 – Computer Graphics

1. SE 313 – Computer Graphics Lecture 8: Transformations and Projections Lecturer: GazihanAlankuş

2. Plan for Today • Post-exam talk • Revisit transformations • Projections

3. Exam Talk • Go over exam questions

4. Transformations (summary) • Three types of linear transformations • Translation (point-vector addition) • Rotation (3x3 matrix multiplication) • Scale (vector-scalar multiplication)

5. Transformations (summary) • Three types of linear transformations • Translation (point-vector addition) • Rotation (3x3 matrix multiplication) • Scale (vector-scalar multiplication) • Cannot combine these operations in one type of operation • Convert them to one type of operation (not possible unless you use homogeneous coordinates)

6. Transformations (summary) • Homogeneous coordinates enable us to represent translation, rotation and scale using 4x4 matrix multiplications. • This way we can combine them easily by multiplying matrices together. The resulting matrix is another transformation.

7. Transformations (summary) • 4x4 matrices that are combinations of translation, rotation and scale Rotation and scale Translation 0 0 0 1

8. Transformations (summary) • You can read the local coordinate frame from 4x4 transformation matrices Rotation and scale Translation The x, y and z axes of thelocal frame Where in the world the local frame’s origin is 0 0 0 1

9. Transformations (summary) • Intuitive understanding of transformations • Local-to-world: insert new transformations near the wall (world) • World-to-local: insert new transformations near the object

10. Transformations (summary) • Quaternions: data structure for rotation • Useful for animations • Ways of representing rotations One axis, one angle 3x3 matrix Quaternion Three angles (euler angles) Best interpolation (slerp) Great-looking animations

11. Plan for Today • Post-exam talk • Revisit transformations • Projections

12. Projection • Projections from 3D to 2D • Taking pictures of the virtual world [Images are borrowed from http://db-in.com]

13. Projection Types • Perspective projection • Just like our eyes and cameras • Orthographic projection • Architectural drawing with no distance distortion

14. Orthographic vs. Perspective Projection [Images are borrowed from http://db-in.com]

15. Orthographic Projection • Get the 3D world, compress it on a 2D paper [engineeringtraining.tpub.com]

16. Orthographic Projection • Great for isometric games (Starcraft, Diablo I-II) • No depth sensation

17. Orthographic Projection in Blender • Select the camera • The viewport is defined by the render output size • Camera has • Scale • Start and end clipping distances

18. Perspective Projection • Take the picture of the world from a single point

19. Perspective Projection • What parameters do I need?

20. Perspective Projection

21. Perspective Projection • How do you do it mathematically? • Also using a 4x4 matrix [songho.ca]

22. Perspective Projection • Let’s try to make sense of it very simply 0 0 0 0 0 0 Translating in z 0 0 -1 0 Output’s w depends on input’s z The further it is in z, the smaller it will get

23. Perspective Projection • What that matrix does

24. Perspective Projection in Blender • Select the camera • The viewport is defined by the render output size • Camera has • Field of view angle • Start and end clipping distances

25. Perspective vs Orthographic Projection Fov=60◦, distance = 1 Fov=30◦, distance = 3 Perspective Fov=10◦, distance = 5 Fov=0◦, distance = Orthographic

26. Perspective vs Orthographic Projection • Orthographic camera is a perspective camera where the camera is at the infinity and the field of view angle is zero

27. Perspective vs Orthographic Projection • In this transition, the size of the arrow in the image stays the same • This is also "called the “dolly-zoom”,“Hitchcock zoom”, or “vertigo effect” • Demonstration in Unity and sample scenes from movies

28. For next week • No homework • Study what we learned today, there will be a quiz • Next week, a part of the lab will be about projection