## Description

### Table of contents:

Cartesian Coordinate Systems

1D Mathematics

2D Cartesian Space

3D Cartesian Space

Odds and ends

Vectors

Vector — mathematical definition and other boring stuff

Vector — a geometric definition

Specifying vectors using Cartesian coordinates

Vectors vs. points

Negating a vector

Vector multiplication by a scalar

Vector addition and subtraction

Vector magnitude (length)

Unit vectors

The distance formula

Vector dot product

Vector cross product

Linear algebra identities

Multiple Coordinate Spaces

Why multiple coordinate spaces?

Some useful coordinate spaces

Coordinate space transformations

Nested coordinate spaces

In defense of upright space

Introduction to Matrices

Matrix — a mathematical definition

Matrix — a geometric interpretation

The bigger picture of linear algebra

Matrices and Linear Transformations

Rotation

Scale

Orthographic projection

Reection

Shearing

Combining transformations

Classes of transformations

More on Matrices

Determinant of a matrix

Inverse of a matrix

Orthogonal matrices

4 x 4 homogeneous matrices

4 x 4 matrices and perspective projection

Polar Coordinate Systems

2D Polar Space

Why would anybody use Polar coordinates?

3D Polar Space

Using polar coordinates to specify vectors

Rotation in Three Dimensions

What exactly is “orientation?”

Matrix form

Euler angles

Axis-angle and exponential map representations

Quaternions

Comparison of methods

Converting between representations

Geometric Primitives

Representation techniques

Lines and rays

Spheres and circles

Bounding boxes

Planes

Triangles

Polygons

Mathematical Topics from 3D Graphics

How graphics works

Viewing in 3D

Coordinate spaces

Polygon meshes

Texture mapping

The standard local lighting model

Light sources

Skeletal animation

Bump mapping

The real-time graphics pipeline

Some HLSL examples

Further reading

Mechanics 1: Linear Kinematics and Calculus

Overview and other expectation-reducing remarks

Basic quantities and units

Average velocity

Instantaneous velocity and the derivative

Acceleration

Motion under constant acceleration

Acceleration and the integral

Uniform circular motion

Mechanics 2: Linear and Rotational Dynamics

Newton’s three laws

Some simple force laws

Momentum

Impulsive forces and collisions

Rotational dynamics

Real-time rigid body simulators

Suggested reading

Curves in 3D

Parametric polynomial curves

Polynomial interpolation

Hermite curves

Bezier curves

Subdivision

Splines

Hermite and Bezier splines

Continuity

Automatic tangent control

Afterword

What next?

Appendix A: Geometric Tests

Appendix B: Answers to the Exercises

Bibliography

Index

Exercises appear at the end of each chapter.

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