1. Transformations: Projection CS 445/645 Introduction to Computer Graphics David Luebke, Spring 2003

2. David Luebke 2 1/24/2016 Admin ● Assn 1 due…

3. David Luebke 3 1/24/2016 Admin ● Assn 1 due… now ■ Any issues?

4. David Luebke 4 1/24/2016 Recap: 3-D Rotation Matrices ● An arbitrary rotation about A can be formed by compositing several canonical rotations together ● We can express each rotation as a matrix ● Compositing transforms == multiplying matrices ● Thus we can express the final rotation as the product of canonical rotation matrices ● Thus we can express the final rotation with a single matrix!

5. David Luebke 5 1/24/2016 Recap: Compositing Matrices ● We have the following matrices: p: The point to be rotated about A by  R y  : Rotate about Y by  R x  : Rotate about X by  R z  : Rotate about Z by  R x  -1 : Undo rotation about X by  R y  -1 : Undo rotation about Y by  ● Thus: p’ = R y  -1 R x  -1 R z  R x  R y  p

6. David Luebke 6 1/24/2016 Recap: Rotation Matrices ● Rotation matrix is orthogonal ■ Columns/rows linearly independent ■ Columns/rows sum to 1 ● The inverse of an orthogonal matrix is its transpose:

7. David Luebke 7 1/24/2016 Recap: Homogeneous Coordinates ● Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector (Note that typically w = 1 in object coordinates)

8. David Luebke 8 1/24/2016 Recap: Homogeneous Coordinates ● Homogeneous coordinates seem unintuitive, but they make graphics operations much easier ● Our transformation matrices are now 4x4:

9. David Luebke 9 1/24/2016 Recap: Homogeneous Coordinates ● Homogeneous coordinates seem unintuitive, but they make graphics operations much easier ● Our transformation matrices are now 4x4:

10. David Luebke 10 1/24/2016 Recap: Homogeneous Coordinates ● Homogeneous coordinates seem unintuitive, but they make graphics operations much easier ● Our transformation matrices are now 4x4:

11. David Luebke 11 1/24/2016 Recap: Homogeneous Coordinates ● Homogeneous coordinates seem unintuitive, but they make graphics operations much easier ● Our transformation matrices are now 4x4:

12. David Luebke 12 1/24/2016 Recap: Homogeneous Coordinates ● Homogeneous coordinates seem unintuitive, but they make graphics operations much easier ● Our transformation matrices are now 4x4:

13. David Luebke 13 1/24/2016 Recap: Translation Matrices ● Now that we can represent translation as a matrix, we can composite it with other transformations ● Ex: rotate 90° about X, then 10 units down Z:

14. David Luebke 14 1/24/2016 Recap: Translation Matrices ● Now that we can represent translation as a matrix, we can composite it with other transformations ● Ex: rotate 90° about X, then 10 units down Z:

15. David Luebke 15 1/24/2016 Recap: Translation Matrices ● Now that we can represent translation as a matrix, we can composite it with other transformations ● Ex: rotate 90° about X, then 10 units down Z:

16. David Luebke 16 1/24/2016 Recap: Translation Matrices ● Now that we can represent translation as a matrix, we can composite it with other transformations ● Ex: rotate 90° about X, then 10 units down Z:

17. David Luebke 17 1/24/2016 More On Homogeneous Coords ● What effect does the following matrix have? ● Conceptually, the fourth coordinate w is a bit like a scale factor

18. David Luebke 18 1/24/2016 More On Homogeneous Coords ● Intuitively: ■ The w coordinate of a homogeneous point is typically 1 ■ Decreasing w makes the point “bigger”, meaning further from the origin ■ Homogeneous points with w = 0 are thus “points at infinity”, meaning infinitely far away in some direction. (What direction?) ■ To help illustrate this, imagine subtracting two homogeneous points

19. David Luebke 19 1/24/2016 Perspective Projection ● In the real world, objects exhibit perspective foreshortening: distant objects appear smaller ● The basic situation:

20. David Luebke 20 1/24/2016 Perspective Projection ● When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world: How tall should this bunny be?

21. David Luebke 21 1/24/2016 Perspective Projection ● The geometry of the situation is that of similar triangles. View from above: ● What is x’? P (x, y, z)X Z View plane d (0,0,0) x’ = ?

22. David Luebke 22 1/24/2016 Perspective Projection ● Desired result for a point [x, y, z, 1] T projected onto the view plane: ● What could a matrix look like to do this?

23. David Luebke 23 1/24/2016 A Perspective Projection Matrix ● Answer:

24. David Luebke 24 1/24/2016 A Perspective Projection Matrix ● Example: ● Or, in 3-D coordinates:

25. David Luebke 25 1/24/2016 A Perspective Projection Matrix ● OpenGL’s gluPerspective() command generates a slightly more complicated matrix: ■ Can you figure out what this matrix does?

26. David Luebke 26 1/24/2016 Projection Matrices ● Now that we can express perspective foreshortening as a matrix, we can composite it onto our other matrices with the usual matrix multiplication ● End result: can create a single matrix encapsulating modeling, viewing, and projection transforms ■ Though you will recall that in practice OpenGL separates the modelview from projection matrix (why?)