Math end algebra. Vector

MATH & ALGEBRA

VECTOR

Length

Pythagorean Formula
|V| = sqrt(x2 + y2)

Addition

A = (1, 2)
B = (4, 0)
A + B = (1+4,

Subtraction

A = (1, 2)
B = (4, 0)
A - B = A

Scalar Multiplication

A*3 = (3*1, 3*2) = (3, 6)
(unit vector = divide the

Dot Product

A = (Ax, Ay, Az) B = (Bx, By, Bz)
A·B =

Cross Product

AxB = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)

Real world examples

In which direction should the missile be fired to hit the target?
Is the enemy visible

Solutions

Solutions have been done by many before.
Know the basics to find

SPACES

Spaces

Euclidean space using Cartesian coordinates. (X, Y and Z)
Local/Model Space
World Space
View/Camera

Left- and right-handed systems

MATRICES AND SPACES

ENTER THE MATRIX

Matrices

Matrix = Transformation placeholder
So again:
Local/Model matrix
World matrix
View/Camera matrix
WVP = world *

Classes/Utils

Matrix3D
Matrix3DUtils
Matrix4x4

TRANSFORMATIONS

Linear transformation

Translation

Linear transformation

Scale

Linear transformation

Skew

Linear transformation

Eulers
Quaternions
Avoids gimbal lock
Slerp (Smooth interpolated rotation)
Matrix – memory intensive

Rotation

Multi linear transformation

Stack of matrices
Apply all at once to an object
The

Nonlinear transformations

Sin curve displacement
Warp

PROJECTIONS

Converting a three-dimensional graphics object
or scene into two dimensions

Most common projections

GRAPHICS PIPELINE

Programmable pipeline

Illustration from db-in.com

Stages overview

Post-processing
Display on screen
or readback: Render to buffer and retrieve values.

SHADERS

The method to render an object.

About shaders

Small programs that runs on the GPU.
Most shader languages are

Low level shading language

Assembly language
ARB (GPU)
AGAL (Adobe Graphics Assembly Language)

!!ARBfp1.0
TEMP

High level shading languages

HLSL – DirectX API
Cg – NVIDIA
GLSL – OpenGL
ShaderLab

Vertex shader

VS or VSH
Executed at each vertex
Transform between coordinate systems
Lighting
Defines

Geometry Shader

Dynamic creation of geometry on the GPU
Only Shader Model 4.0
Direct3D

Fragment Shader

FSH
Processed at each visible fragment
Fragment != Pixel
Handles bump effects, shadows

Texture objects

Texels
Power of Two (POT)  2, 4,…512, 1024 pixels
Flipped pixel order

Texture Filtering

Fixing artifacts
Texture magnification/minification
Mipmapping
Different techniques:

Let’s have a look at the WegGL implementation (click on image)

three.js

Cubemap texture

3D texture
Skybox
Reflections
Environment map

Shader tool examples

Shader Toy – WebGL
MeShade – WebGL
PixelBender3D – Molehill
Node

Interior mapping

Animations, Skin and Bones

Tweens
Animation controllers Blending Mixing/Additive 
Vertex animations in shader
Procedurally animating

Animations in Away3D Broomstick

Materials

Material is the collection of properties applied to an object.
Shaders is

Some ingredients:

Color
Diffuse: base color
Ambient: color of ambient light (shadowed parts). Mostly

Example

Unitys Normal Shader Family

Lighting

Uses normals
Directional/point-lights
Material settings to decide final color.
Lighting is computed at each

Lambert shading

Real-time shadows

Quality and performance

Non realtime-shadows fastest!
Shadow map resolution
Number of lights

Example in Unity

Special effects

Effects
Color correction
Postprocessing stage / GPU
LDR/HDR, Tone mapping

Physics

Very simple physics demo

Frameworks

Goal: Games, experimental, Vizualisation?
Reach: Plugin? Multiple platforms/screens?
Cost: Open source? Licenced?
Support: Large

Unity3D

Boo, C# and JavaScript
Plugin
Great and simple IDE
Competent and mature framework
Pro version

Flash/Molehill

Actionscript
Plugin
3D content always under the DisplayList
All the other stuff in

Flash 3D Engines

Optimizing

Profiling memory usage, cleanup/destroy
Object Pooling! polygonal lab
Take control of rendering pipeline
Compression/Model

WebGL

Javascript
No plugin
Open / Royalty-free
Not available in all browsers yet
Frameworks in early

WebGL Frameworks

Jellyfish

Aleksandar Rodic

Particles

alteredqualia.com

Hello Racer

HelloEnjoy™

Clouds

Mr Doob

WebGL vs. Molehill APIs

HTML5 vs. Plugin.
WebGL will probably run in iOS

Debugging

Profiling CPU
FlashPreloadProfiler
Profiling GPU
Pix for windows
Intel® Graphics Performance Analyzers

3D Model filetypes

Learning tips

Random interesting topics

Random interesting topics

Level of detail

Octree, BSP Tree, Portals and Sectors

Random interesting topics

Global illumination / Ambient occlusion

Random interesting topics

Raytracing/Raycasting/Raymarching

Random interesting topics

Some useful resources

Books and papers

Away3D 3.6 essentials
Mathematics for Game Developer by Christopher Tremblay
Mathematics