Mathematics for Game Development

11.

Specialized Topics

11.1.

Procedural Noise Functions

11.1.1.

Noise Fundamentals

11.1.1.1.

Deterministic Randomness

11.1.1.2.

Spatial Coherence

11.1.1.3.

Frequency and Amplitude

11.1.2.

11.1.2.1.

Grid-Based Approach

11.1.2.2.

Gradient Vectors

11.1.2.3.

Interpolation Methods

11.1.2.4.

Octave Combination

11.1.3.

11.1.3.1.

11.1.3.2.

Gradient Selection

11.1.3.3.

Performance Advantages

11.1.4.

Noise Variations

11.1.4.1.

11.1.4.2.

11.1.4.3.

11.1.4.4.

11.1.5.

Noise Applications

11.1.5.1.

Terrain Generation

11.1.5.1.1.

11.1.5.1.2.

Erosion Simulation

11.1.5.1.3.

Biome Distribution

11.1.5.2.

Texture Synthesis

11.1.5.2.1.

Surface Details

11.1.5.2.2.

Material Variation

11.1.5.2.3.

Weathering Effects

11.1.5.3.

Environmental Effects

11.1.5.3.1.

Cloud Generation

11.1.5.3.2.

Water Simulation

11.1.5.3.3.

Particle Systems

11.2.

Graphics Pipeline Mathematics

11.2.1.

Coordinate System Transformations

11.2.1.1.

Model Coordinates

11.2.1.2.

World Coordinates

11.2.1.3.

View Coordinates

11.2.1.4.

Clip Coordinates

11.2.1.5.

Screen Coordinates

11.2.2.

Model-View-Projection Pipeline

11.2.2.1.

11.2.2.1.1.

Object Transformation

11.2.2.1.2.

Local to World Space

11.2.2.2.

11.2.2.2.1.

Camera Transformation

11.2.2.2.2.

World to View Space

11.2.2.2.3.

11.2.2.3.

Projection Matrix

11.2.2.3.1.

Perspective Projection

11.2.2.3.2.

Orthographic Projection

11.2.2.3.3.

Projection Parameters

11.2.2.4.

MVP Matrix Combination

11.2.2.4.1.

Matrix Multiplication Order

11.2.2.4.2.

Vertex Transformation

11.2.3.

Clipping and Culling

11.2.3.1.

11.2.3.2.

Clipping Planes

11.2.3.3.

Back-Face Culling

11.2.3.4.

Occlusion Culling

11.2.4.

Viewport Transformation

11.2.4.1.

Screen Space Conversion

11.2.4.2.

11.2.4.3.

Pixel Coordinates

11.3.

Lighting and Shading Mathematics

11.3.1.

11.3.1.1.

11.3.1.2.

Directional Lights

11.3.1.3.

11.3.1.4.

11.3.2.

Lighting Calculations

11.3.2.1.

Ambient Lighting

11.3.2.1.1.

Global Illumination Approximation

11.3.2.2.

Diffuse Lighting

11.3.2.2.1.

11.3.2.2.2.

Surface Normal Importance

11.3.2.3.

Specular Lighting

11.3.2.3.1.

11.3.2.3.2.

Blinn-Phong Model

11.3.2.3.3.

Reflection Vectors

11.3.3.

Advanced Lighting

11.3.3.1.

Attenuation Models

11.3.3.1.1.

Distance-Based Falloff

11.3.3.1.2.

Quadratic Attenuation

11.3.3.2.

Shadow Calculations

11.3.3.2.1.

11.3.3.2.2.

11.3.3.3.

Global Illumination

11.3.3.3.1.

11.3.3.3.2.

11.3.4.

Shader Mathematics

11.3.4.1.

Vertex Shader Operations

11.3.4.1.1.

Position Transformation

11.3.4.1.2.

Normal Transformation

11.3.4.1.3.

Texture Coordinate Manipulation

11.3.4.2.

Fragment Shader Operations

11.3.4.2.1.

Color Calculations

11.3.4.2.2.

Texture Sampling

11.3.4.2.3.

11.3.4.3.

Mathematical Functions in Shaders

11.3.4.3.1.

Built-in Functions

11.3.4.3.2.

Custom Mathematical Operations

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1. Foundational Mathematical Concepts