Matthew Christian

About MeIntroduction to Linear AlgebraVectorsMatricesQuaternionsLinks

StudentApplied Mathematics and Computer Science:

Software Development at UW-StoutDegree

Associates Degree in Computer Programming from Northcentral Technical College

GamerBeaten over 140 games (Yes I have a list)Independent Game Developer (spare time) for

5 yearsXNA developer for around a year and a half

DefinitionThe part of algebra that deals with the theory

of linear equations and linear transformationsIn which the specific properties of vector

spaces are studied (including matrices)

This is NOT about ‘Linear Algebra’, it’s about the gaming version of ‘Linear Algebra’

DefinitionsA variable quantity that can be resolved into

componentsA straight line segment whose length is

magnitude and whose orientation in space is direction

Vectors are simple row-based data structuresIn XNA:

Microsoft.XNA.FrameworkVector2, Vector3, Vector4public Vector2 ( float x, float y );

Let P be a point at (2,2)

Then the vector p can be described as:

p = [2, 2]

Ex: [2,1] , [1,3]

Vector AdditionAdd terms in similar positions

Vector “Subtraction”Remember, vectors represent directionsHow to subtract direction? Add negative

directionScalar-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

Dot ProductHelps determine the angle between 2 vectors

Cross ProductCreates another vector

perpendicular to the other two vectors (normal) (3D)

NormalizingMagnitude (length)

Storing values (positions)Directions (move direction, collision

direction)

Demo(s)Simple VectorsVector Collision

DefinitionA 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 columnsNxm matrix is a matrix with n rows and m

columns

Square MatrixN-rows, N-columns

Main DiagonalRuns 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 MatrixThe matrix equivalent of multiplying by 1; 1’s

across the main diagonal with zero’s elsewhere (nxn sized)

3D Rendering is possible BECAUSE of matrices

ModelViewProjection MatrixModel Matrix – Matrix describing the

position/rotation/scale of your objectOrder is important (multiply in order of operations)

View Matrix – Camera position, target, up direction (orientation)

Projection Matrix – View frustum ‘squished’ (your monitor doesn’t display ‘3D’)

Math DemosTransformations!

In XNAMicrosoft.XNA.FrameworkMatrix4x4 matrix (M11 – M44)

DemosMatrixTransformationsCamera Demo (from Tutorials)

Quaternions are compact descriptions of rotations…

Quaternions DON’T Prevent Gimbal LockMatrices use Euler numbers to calculate

rotations which ‘cancels’ a directionAfter calculating, you can only rotate on the Z-

Axis

In all honesty, I’m still researching it!

(X, Y, Z, W)(X, Y, Z) is the axis to do rotations about(W) is the amount to rotate about that axis

Arbitrary AxisNot global axis

DemoQuaternion Camera

http://www.insidegamer.org/XnaTutorials.aspxMy tutorials (specifically Tutorial 4)

http://www.ziggyware.com/readarticle.php?article_id=54Specifically about Vectors in XNA

http://hyperphysics.phy-astr.gsu.edu/hbase/vect.htmlSome Vector operations

http://geekswithblogs.net/CodeBlogMy (seldom updated) Blog