Matthew Christian

1
Basics of 3D Math in Games

• Matthew Christian

2
Overview

• About Me
• Introduction to Linear Algebra
• Vectors
• Matrices
• Quaternions
• Links

3
About Me

• Student
• Applied Mathematics and Computer Science
  Software Development at UW-Stout
• Degree
• Associates Degree in Computer Programming from
  Northcentral Technical College
• Gamer
• Beaten over 140 games (Yes I have a list)
• Independent Game Developer (spare time) for 5
  years
• XNA developer for around a year and a half

4
Intro to Linear Algebra

• Definition
• The part of algebra that deals with the theory of
  linear equations and linear transformations
• In which the specific properties of vector spaces
  are studied (including matrices)
• This is NOT about Linear Algebra, its about
  the gaming version of Linear Algebra

5
Vectors

• Definitions
• A variable quantity that can be resolved into
  components
• A straight line segment whose length is magnitude
  and whose orientation in space is direction
• Vectors are simple row-based data structures
• In XNA
• Microsoft.XNA.Framework
• Vector2, Vector3, Vector4
• public Vector2 ( float x, float y )

6
Cartesian Example
Let P be a point at (2,2) Then the vector p can
be described as p 2, 2 Ex 2,1 , 1,3
7
Vector Operations

• Vector Addition
• Add terms in similar positions
• Vector Subtraction
• Remember, vectors represent directions
• How to subtract direction? Add negative
  direction
• Scalar-Vector Multiplication
• Scaling a Vector up or down is easy, multiply
  each element by the scalar
• Similar for Division (multiply by scalar
  fraction)
• Vector Matrix Multiplication
• See later

8
More Advanced Operations

• Dot Product
• Helps determine the angle between 2 vectors
• Cross Product
• Creates another vector perpendicular to the
  other two vectors (normal) (3D)
• Normalizing
• Magnitude (length)

9
Basic Uses for Vectors

• Storing values (positions)
• Directions (move direction, collision direction)
• Demo(s)
• Simple Vectors
• Vector Collision

10
Matrices

• Definition
• A rectangular array of quantities set out by
  rows and columns, treated as a single element and
  manipulated accordingly
• For us programmers,
• Multi-dimensional arrays
• A column is a 3x3 matrix if it has 3 rows and 3
  columns
• Nxm matrix is a matrix with n rows and m columns

11
Anatomy of a Matrix

• Square Matrix
• N-rows, N-columns
• Main Diagonal
• Runs from upper left corner down (includes
  non-square matrices)
• Diagonal Matrix
• Matrix where all entries outside of the main
  diagonal are zero (main diagonal entries can be
  zero)
• Identity Matrix
• The matrix equivalent of multiplying by 1 1s
  across the main diagonal with zeros elsewhere
  (nxn sized)

12
3D Rendering

• 3D Rendering is possible BECAUSE of matrices
• ModelViewProjection Matrix
• Model Matrix Matrix describing the
  position/rotation/scale of your object
• Order is important (multiply in order of
  operations)
• View Matrix Camera position, target, up
  direction (orientation)
• Projection Matrix View frustum squished (your
  monitor doesnt display 3D)

13
Uses for Matrices

• Math Demos
• Transformations!
• In XNA
• Microsoft.XNA.Framework
• Matrix
• 4x4 matrix (M11 M44)
• Demos
• MatrixTransformations
• Camera Demo (from Tutorials)

14
Quaternions

• Quaternions are compact descriptions of
  rotations
• Quaternions DONT Prevent Gimbal Lock
• Matrices use Euler numbers to calculate rotations
  which cancels a direction
• After calculating, you can only rotate on the
  Z-Axis
• In all honesty, Im still researching it!

15
Quaternions (cont'd)

• (X, Y, Z, W)
• (X, Y, Z) is the axis to do rotations about
• (W) is the amount to rotate about that axis
• Arbitrary Axis
• Not global axis

16
Uses for Quaternions

• Demo
• Quaternion Camera

17
Questions? Links

• http//www.insidegamer.org/XnaTutorials.aspx
• My tutorials (specifically Tutorial 4)
• http//www.ziggyware.com/readarticle.php?article_i
  d54
• Specifically about Vectors in XNA
• http//hyperphysics.phy-astr.gsu.edu/hbase/vect.ht
  ml
• Some Vector operations
• http//geekswithblogs.net/CodeBlog
• My (seldom updated) Blog