1. Video Game Rendering Mathematics
Zack Booth Simpson

2. From unrealtechnology.com
How to we model a body?
Break each object up into lots of triangles. Each “face” is connected to a mesh of “vertices”
Make each part of a “body” rigid
Connect the rigid parts together with joints
Let the joints rotate
From unrealtechnology.com

3. Rendering Meshes
Each vertex is positioned by an artist. The programmers get this vertex list and tell the computer to connect the dots.
But the vertices are in 3D?!
How do we get from a 3D model to a 2D view on the screen?
Create a virtual camera

4. Pretend Cameras What does the camera see of this box?
Two problems to solve:
Relative position and orientation of the box versus the camera
The perspective distortion

5. Perspective: Things further away appear smaller
Create a pretend “projection plane” that represents the computer screen

6. Exercise: The Perspectometer

7. Answers: The Perspectometer
The length of the line on the board is 3 feet.
The units are feet, because the distance from your eye to the board was measured in feet.
Note that the length of your arm and the apparent height of the line were both measured in inches.

8. Position and Orientation
Meet “vector”
From Latin “vehere” = “to carry” same root as “vehicle”
Each object in the world is placed and oriented.
Vectors let us get from place to place and orient ourselves.
Where are the vertices of the box from the point of view of the camera?
(Never mind perspective for now)

9. Translate = Move without rotate
Relative position is easy
Vector addition and subtraction!
a + b - c = d
d is relative to the camera… we say “d is the position of the vertex in camera space”

10. Rotation Now things get harder
From the camera’s point of view the box is rotated
From the box’s point of view the camera is rotated
Coping with differing points of view is always a challenge!

15. Who’s point of view is right?

16. Both / Neither side is right
We need a way to measure each vector using the other one as the unit ruler.
In other words: we need tools to convert from one point of view to the other.

17. Projection a/b from b’s perspective; b/a from a’s perspective
Remember always: “a/b” means “measure a against b” or “using b as the ruler, how long is a?”
It is easy to forget/get confused about this… I wish a teacher had beaten this into my head with a stick!

18. Oh boy, here we go again…

19. Projection from different directions
Each side can be projected onto the other.

20. As a is rotated away from b, the projection diminishes until at 90 degrees it is zero
Perpendicular vectors can not measure each other; they have zero relative projections!

21. Projection with Cosine
We don’t want no stinking cosines – they’re a pain!
Here’s a great trick…

22. Ever seen this lovely proof of the Pythagorean theorem?

23. Rectangular Version of the Pythagorean Theorem
Dot product graphic removed pending publication
Cool! We can compute the projection with just two multiplies and add! Games have millions of vertices so this is very important! Considering speed is the difference between mathematics and computation!

24. Projection with the dot product from b’s point of view
TADA! p can be computed without angles or cosines!

25. More Dot Product Goodness
Dot product sign tells you about the relationship of the vectors.
Same general direction
Orthogonal
Opposite directions

26. Exercise: Dot Products

27. Answers: Dot Products

28. Back to Rotations Ok, so now we can project a vector onto another
What’s this got to do with rotations again?
Represent the camera’s POV as aligned vectors. Call these “basis vectors” as in “base”
The projection onto these vectors gives us the rotation we want! i.e. the box from the POV of camera

29. Basis Vector Demo Run demo app
Skew, rotation, scale, reflection, projection
Orthogonal basis vectors create a rotation or reflection

30. The Matrix A cool way to organize basis vectors: The Matrix
The default basis vectors are one unit in each direction.
Write each basis vector as a column in the matrix
Write a block of numbers but visualize vectors!
The matrix encodes any “linear” transformation

31. The Power of The Matrix

32. The Matrix as a Teleporter
The matrix is a “teleportation machine” which moves points from one space to another

33. A Teleporter inside a Teleporter?!
What happens if you put a teleporter inside a teleporter?
Compounding teleportations!

34. Matrix Concatenation
Map the basis vectors of one space into the another

35. Matrix Concatenation Using Dot product

36. Matrix Concatenation Using Dot product

37. Matrix Concatenation Using Dot product

38. Matrix Concatenation Using Dot product

39. Exercise: Matrix Arithmetic

40. Answers: Matrix Arithmetic

41. Rotate, Scale, and Translate?
So now we can rotate, scale, etc. but what ever happened to translations?
The ugly way
Would be nice to matrixify™ it!

42. Stuffin’ the Matrix Make up a pretend dimension for the translation
Hardwire “1” unit of this extra dimension in all vertices to be transformed.
Where is this extra dimension exactly?
Who cares! It works! Math is so cool

43. Moving to 3D Just add a dimension again. Now matrices are 4x4
Right-hand rule
From math.montana.edu

44. Exercise: Matrix Transforms

45. Answers: Matrix Transformations

46. Game: Matrix Mine Sweeper
There’s a bomb in a house
There’s two rounds
You must move the bomb with one of the three given matrices in each round
You can not go off the map
Plan ahead!!

47. Game: Matrix Mine Sweeper
There’s a bomb in a house
There’s two rounds
You must move the bomb with one of the three given matrices in each round
You can not go off the map
Plan ahead!!
GO!

48. Answers: Matrix Mine Sweeper Round 1

49. Game: Matrix Mine Sweeper
Round 2. GO!!

50. Answers: Matrix Mine Sweeper Round 2

51. Answers: Matrix Mine Sweeper Bonus Round!!
If you are outside a building, you’re a hero!

52. Answers: Matrix Mine Sweeper Bonus Round!!
If you are outside a building, you’re a hero!
If you are inside, this is your last chance! You are almost dead! Quick! Make up your own matrix!
You have 20 seconds! GO!

53. BOOM!
From

54. A final desperate solution! The zero matrix!
Collapse all paths to a singularity!