1. and an introduction to matrices
Coordinate Systems
and an introduction to matrices
Jeff Chastine

2. The Local Coordinate System
Sometimes called “Object Space”
It’s the coordinate system the model was made in
Jeff Chastine

3. The Local Coordinate System
Sometimes called “Object Space”
It’s the coordinate system the model was made in
(0, 0, 0)
Jeff Chastine

4. The World SPACE
The coordinate system of the virtual environment
(619, 10, 628)
Jeff Chastine

5. (619, 10, 628)
Jeff Chastine

6. Question How did get the monster positioned correctly in the world?
Let’s come back to that…
Jeff Chastine

7. Camera Space
It’s all relative to the camera…
Jeff Chastine

8. Camera Space
It’s all relative to the camera… and the camera never moves!
(0, 0, -10)
Jeff Chastine

9. The Big Picture
How to we get from space to space? ? ?
Jeff Chastine

10. The Big Picture M ? How to we get from space to space? For every model
Have a (M)odel matrix!
Transforms from object to world space M ?
Jeff Chastine

11. The Big Picture M V How to we get from space to space?
To put in camera space
Have a (V)iew matrix
Usually need only one of these M V
Jeff Chastine

12. The Big Picture M V MV How to we get from space to space?
The ModelView matrix
Sometimes these are combined into one matrix
Usually keep them separate for convenience M V MV
Jeff Chastine

13. Matrix - What?
A mathematical structure that can:
Translate (a.k.a. move)
Rotate
Scale
Usually a 4x4 array of values
Idea: multiply each point by a matrix to get the new point
Your graphics card eats matrices for breakfast
The Identity Matrix
Jeff Chastine

14. Back to The Big Picture M
If you multiply a matrix by a matrix, you get a matrix!
How might we make the model matrix? M
Jeff Chastine

15. Back to The Big Picture M Translation matrix T Rotation matrix R1
Scale matrix S
If you multiply a matrix by a matrix, you get a matrix!
How might we make the model matrix? M
Jeff Chastine

16. Back to The Big Picture M T * R1 * R2 * S = M Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
If you multiply a matrix by a matrix, you get a matrix!
How might we make the model matrix? M
T * R1 * R2 * S = M
Jeff Chastine

17. Matrix Order (an angry vertex) Multiply left to right
Results are drastically different
(an angry vertex)
Jeff Chastine

18. Matrix Order Multiply left to right Results are drastically different
Order of operations
Rotate 45°
Jeff Chastine

19. Matrix Order Multiply left to right Results are drastically different
Order of operations
Rotate 45°
Translate 10 units
Jeff Chastine

20. Matrix Order Multiply left to right Results are drastically different
Order of operations
Rotate 45°
Translate 10 units
before
after
Jeff Chastine

21. Matrix Order Multiply left to right Results are drastically different
Order of operations
Jeff Chastine

22. Matrix Order Multiply left to right Results are drastically different
Order of operations
Translate 10 units
Jeff Chastine

23. Matrix Order Multiply left to right Results are drastically different
Order of operations
Translate 10 units
Rotate 45°
Jeff Chastine

24. Matrix Order Multiply left to right Results are drastically different
Order of operations
Translate 10 units
Rotate 45°
after
before
Jeff Chastine

25. Back to The Big Picture M T * R1 * R2 * S = M Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
If you multiply a matrix by a matrix, you get a matrix!
How might we make the model matrix? M
T * R1 * R2 * S = M
Backwards
Jeff Chastine

26. Back to The Big Picture M S * R1 * R2 * T = M Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
If you multiply a matrix by a matrix, you get a matrix!
How might we make the model matrix? M
S * R1 * R2 * T = M
Jeff Chastine

27. The (P)rojection Matrix
Projects from 3D into 2D
Two kinds:
Orthographic: depth doesn’t matter, parallel remains parallel
Perspective: Used to give depth to the scene (a vanishing point)
End result: Normalized Device Coordinates (NDCs between -1.0 and +1.0)
Jeff Chastine

28. Orthographic vs. Perspective
Jeff Chastine

29. An Old Vertex Shader Originally we passed using NDCs (-1 to +1)
in vec4 vPosition; // The vertex in NDC void main () { gl_Position = vPosition; }
Originally we passed using NDCs (-1 to +1)
Jeff Chastine

30. A Better Vertex Shader New position in NDC Original (local) position
in vec4 vPosition; // The vertex in the local coordinate system uniform mat4 mM; // The matrix for the pose of the model uniform mat4 mV; // The matrix for the pose of the camera uniform mat4 mP; // The projection matrix (perspective) void main () { gl_Position = mP*mV*mM*vPosition; }
New position in NDC
Original (local) position
Jeff Chastine

31. SMILE – It’s the END!
Jeff Chastine