Video Game Rendering Mathematics Zack Booth Simpson

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

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

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

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

Exercise: The Perspectometer

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.

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)

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”

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!

Who’s point of view is right?

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.

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!

Oh boy, here we go again…

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

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!

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

Ever seen this lovely proof of the Pythagorean theorem?

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!

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

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

Exercise: Dot Products

Answers: Dot Products

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

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

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

The Power of The Matrix

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

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

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

Matrix Concatenation Using Dot product

Matrix Concatenation Using Dot product

Matrix Concatenation Using Dot product

Matrix Concatenation Using Dot product

Exercise: Matrix Arithmetic

Answers: Matrix Arithmetic

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!

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

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

Exercise: Matrix Transforms

Answers: Matrix Transformations

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!!

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!

Answers: Matrix Mine Sweeper Round 1

Game: Matrix Mine Sweeper Round 2. GO!!

Answers: Matrix Mine Sweeper Round 2

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

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!

BOOM! From www.prostunts.net

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