Foundation Game of Engine Development: Volume 1 - Mathematics

Chapter 1: Vectors and Matrices

1.1 Vector Fundamentals
1.2 Basic Vector Operations
1.2.1 Magnitude and Scalar Multiplication
1.2.2 Addition and Subtraction
1.3 Matrix Fundamentals
1.4 Basic Matrix Operations
1.4.1 Addition, Subtraction, and Scalar Multiplication
1.4.2 Matrix Multiplication
1.5 Vector Multiplication
1.5.1 Dot Product
1.5.2 Cross Product
1.5.3 Scalar Triple Product
1.6 Vector Projection
1.7 Matrix Inversion
1.7.1 Identity Matrices
1.7.2 Determinants
1.7.3 Elementary Matrices
1.7.4 Inverse Calculation
1.7.5 Inverses of Small Matrices
Exercises for Chapter 1
Chapter 2: Transforms

2.1 Coordinate Spaces
2.1.1 Transformation Matrices
2.1.2 Orthogonal Transforms
2.1.3 Transform Composition
2.2 Rotations
2.2.1 Rotation About a Coordinate Axis
2.2.2 Rotation About an Arbitrary Axis
2.3 Reflections
2.4 Scales
2.5 Skews
2.6 Homogeneous Coordinates
2.7 Quaternions
2.7.1 Quaternion Fundamentals
2.7.2 Rotations With Quaternions
Exercises for Chapter 2
Chapter 3: Geometry

3.1 Triangle Meshes
3.2 Normal Vectors
3.2.1 Calculating Normal Vectors
3.2.2 Transforming Normal Vectors
3.3 Lines and Rays
3.3.1 Parametric Lines
3.3.2 Distance Between a Point and a Line
3.3.3 Distance Between Two Lines
3.4 Planes
3.4.1 Implicit Planes
3.4.2 Distance Between a Point and a Plane
3.4.3 Reflection Through a Plane
3.4.4 Intersection of a Line and a Plane
3.4.5 Intersection of Three Planes
3.4.6 Intersection of Two Planes
3.4.7 Transforming Planes
3.5 Plücker Coordinates
3.5.1 Implicit Lines
3.5.2 Homogeneous Formulas
3.5.3 Transforming Lines
Exercises for Chapter 3
Chapter 4: Advanced Algebra

4.1 Grassmann Algebra
4.1.1 Wedge Product
4.1.2 Bivectors
4.1.3 Trivectors
4.1.4 Algebraic Structure
4.1.5 Complements
4.1.6 Antivectors
4.1.7 Antiwedge Product
4.2 Projective Geometry
4.2.1 Lines
4.2.2 Planes
4.2.3 Join and Meet
4.2.4 Line Crossing
4.2.5 Plane Distance
4.2.6 Summary and Implementation
4.3 Matrix Inverses
4.4 Geometric Algebra
4.4.1 Geometric Product
4.4.2 Vector Division
4.4.3 Rotors
4.5 Conclusion
Exercises for Chapter 4
Index

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