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By Philip Schneider, Industrial Light + Magic, San Francisco, California, U.S.A. David Eberly, Geometric Tools, Inc., Chapel Hill, North Carolina, U.S.A. ACM President of Geometric Tools, Inc (www.geometrictools.com), a company
that specializes in software development for computer graphics, image analysis, and numerical methods. Previously, he was the Director
of Engineering at Numerical Design Ltd (NDL), the company responsible for the real-time 3D game engine, Netlmmerse. His background includes
a BA in Mathematics from Bloomsburg U, MS and PhD degrees in Mathematics from the U of Colorado at Boulder, and MS and PhD degrees in
computer science from the U of North Carolina at Chapel Hill. He is the author of 3D Game Engine Design, 2e, and 3D Game Engine Architecture,
and co-author of Geometric Tools for Computer Graphics (all published by Morgan Kaufmann).
Do you spend too much time creating the building blocks of your graphics applications or finding and correcting errors? Geometric
Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that
you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition,
intersection determination, separation, and more.
If you have a mathematics degree, this book will save you time and trouble. If
you don't, it will help you achieve things you may feel are out of your reach. Inside, each problem is clearly stated and diagrammed,
and the fully detailed solutions are presented in easy-to-understand pseudocode. You also get the mathematics and geometry background
needed to make optimal use of the solutions, as well as an abundance of reference material contained in a series of appendices.
Audience
Programmers, software engineers, and technical directors whose work involves 2D or 3D computer graphics for filmmaking including special
effects and animation programming, gane development, visualization, medical image analysis, simulation, virtual worlds, and other software
development.
Contents Foreword
Figures
Tables
Preface
Chapter 1 Introduction
1.1 How to Use This Book
1.2 Issues of Numerical Computation
1.2.1
Low-Level Issues
1.2.2 High-Level Issues
1.3 A Summary of the Chapters
Chapter 2 Matrices and Linear Systems
2.1 Introduction
2.1.1 Motivation
2.1.2 Organization
2.1.3 Notational Conventions
2.2 Tuples
2.2.1 Definition
2.2.2
Arithmetic Operations
2.3 Matrices
2.3.1 Notation and Terminology
2.3.2 Transposition
2.3.3 Arithmetic Operations
2.3.4 Matrix Multiplication
2.4 Linear Systems
2.4.1 Linear Equations
2.4.2 Linear Systems in Two Unknowns
2.4.3 General Linear Systems
2.4.4 Row Reductions, Echelon Form, and Rank
2.5 Square Matrices
2.5.1 Diagonal
Matrices
2.5.2 Triangular Matrices
2.5.3 The Determinant
2.5.4 Inverse
2.6 Linear Spaces
2.6.1 Fields
2.6.2 Definition and Properties
2.6.3 Subspaces
2.6.4 Linear Combinations and Span
2.6.5 Linear Independence,
Dimension, and Basis
2.7 Linear Mappings
2.7.1 Mappings in General
2.7.2 Linear Mappings
2.7.3 Matrix Representation
of Linear Mappings
2.7.4 Cramer?s Rule
2.8 Eigenvalues and Eigenvectors
2.9 Euclidean Space
2.9.1 Inner Product
Spaces
2.9.2 Orthogonality and Orthonormal Sets
2.10 Least Squares
Recommended Reading
Chapter 3 Vector Algebra
3.1 Vector Basics
3.1.1 Vector Equivalence
3.1.2 Vector Addition
3.1.3 Vector Subtraction
3.1.4 Vector Scaling
3.1.5 Properties of Vector Addition and Scalar Multiplication
3.2 Vector Space
3.2.1 Span
3.2.2 Linear Independence
3.2.3 Basis, Subspaces, and Dimension
3.2.4 Orientation
3.2.5 Change of Basis
3.2.6 Linear Transformations
3.3 Affine Spaces
3.3.1 Euclidean Geometry
3.3.2 Volume, the Determinant, and the Scalar Triple Product
3.3.3
Frames
3.4 Affine Transformations
3.4.1 Types of Affine Maps
3.4.2 Composition of Affine Maps
3.5 Barycentric Coordinates
and Simplexes
3.5.1 Barycentric Coordinates and Subspaces
3.5.2 Affine Independence
Chapter 4 Matrices, Vector Algebra,
and Transformations
4.1 Introduction
4.2 Matrix Representation of Points and Vectors
4.3 Addition, Subtraction, and Multiplication
4.3.1 Vector Addition and Subtraction
4.3.2 Point and Vector Addition and Subtraction
4.3.3 Subtraction of Points
4.3.4 Scalar Multiplication
4.4 Products of Vectors
4.4.1 Dot Product
4.4.2 Cross Product
4.4.3 Tensor
Product
4.4.4 The ?Perp? Operator and the ?Perp? Dot Product
4.5 Matrix Representation of Affine Transformations
4.6 Change-of-Basis/Frame/Coordinate
System
4.7 Vector Geometry of Affine Transformations
4.7.1 Notation
4.7.2 Translation
4.7.3 Rotation
4.7.4 Scaling
4.7.5 Reflection
4.7.6 Shearing
4.8 Projections
4.8.1 Orthographic
4.8.2 Oblique
4.8.3 Perspective
4.9 Transforming Normal Vectors
Recommended Reading
Chapter 5 Geometric Primitives in 2D
5.1 Linear
Components
5.1.1 Implicit Form
5.1.2 Parametric Form
5.1.3 Converting between Representations
5.2 Triangles
5.3 Rectangles
5.4 Polylines and Polygons
5.5 Quadratic Curves
5.5.1 Circles
5.5.2 Ellipses
5.6 Polynomial
Curves
5.6.1 B'ezier Curves
5.6.2 B-Spline Curves
5.6.3 NURBS Curves
Chapter 6 Distance in 2D
6.1 Point to
Linear Component
6.1.1 Point to Line
6.1.2 Point to Ray
6.1.3 Point to Segment
6.2 Point to Polyline
6.3
Point to Polygon
6.3.1 Point to Triangle
6.3.2 Point to Rectangle
6.3.3 Point to Orthogonal Frustum
6.3.4
Point to Convex Polygon
6.4 Point to Quadratic Curve
6.5 Point to Polynomial Curve
6.6 Linear Components
6.6.1 Line
to Line
6.6.2 Line to Ray
6.6.3 Line to Segment
6.6.4 Ray to Ray
6.6.5 Ray to Segment
6.6.6 Segment
to Segment
6.7 Linear Component to Polyline or Polygon
6.8 Linear Component to Quadratic Curve
6.9 Linear Component to Polynomial
Curve
6.10 GJK Algorithm
6.10.1 Set Operations
6.10.2 Overview of the Algorithm
6.10.3 Alternatives to GJK
Chapter 7 Intersection in 2D
7.1 Linear Components
7.2 Linear Components and Polylines
7.3 Linear Components and Quadratic
Curves
7.3.1 Linear Components and General Quadratic Curves
7.3.2 Linear Components and Circular Components
7.4 Linear
Components and Polynomial Curves
7.4.1 Algebraic Method
7.4.2 Polyline Approximation
7.4.3 Hierarchical Bounding
7.4.4 Monotone Decomposition
7.4.5 Rasterization
7.5 Quadratic Curves
7.5.1 General Quadratic Curves
7.5.2
Circular Components
7.5.3 Ellipses
7.6 Polynomial Curves
7.6.1 Algebraic Method
7.6.2 Polyline Approximation
7.6.3 Hierarchical Bounding
7.6.4 Rasterization
7.7 The Method of Separating Axes
7.7.1 Separation by Projection
onto a Line
7.7.2 Separation of Stationary Convex Polygons
7.7.3 Separation of Moving Convex Polygons
7.7.4 Intersection
Set for Stationary Convex Polygons
7.7.5 Contact Set for Moving Convex Polygons
Chapter 8 Miscellaneous 2D Problems
8.1
Circle through Three Points
8.2 Circle Tangent to Three Lines
8.3 Line Tangent to a Circle at a Given Point
8.4 Line Tangent
to a Circle through a Given Point
8.5 Lines Tangent to Two Circles
8.6 Circle through Two Points with a Given Radius
8.7
Circle through a Point and Tangent to a Line with a Given Radius
8.8 Circles Tangent to Two Lines with a Given Radius
8.9 Circles
through a Point and Tangent to a Circle with a Given Radius
8.10 Circles Tangent to a Line and a Circle with a Given Radius
8.11
Circles Tangent to Two Circles with a Given Radius
8.12 Line Perpendicular to a Given Line through a Given Point
8.13 Line between
and Equidistant to Two Points
8.14 Line Parallel to a Given Line at a Given Distance
8.15 Line Parallel to a Given Line at a
Given Vertical (Horizontal) Distance
8.16 Lines Tangent to a Given Circle and Normal to a Given Line
Chapter 9 Geometric Primitives
in 3D
9.1 Linear Components
9.2 Planar Components
9.2.1 Planes
9.2.2 Coordinate System Relative to a Plane
9.2.3 2D Objects in a Plane
9.3 Polymeshes, Polyhedra, and Polytopes
9.3.1 Vertex-Edge-Face Tables
9.3.2 Connected
Meshes
9.3.3 Manifold Meshes
9.3.4 Closed Meshes
9.3.5 Consistent Ordering
9.3.6 Platonic Solids
9.4
Quadric Surfaces
9.4.1 Three Nonzero Eigenvalues
9.4.2 Two Nonzero Eigenvalues
9.4.3 One Nonzero Eigenvalue
9.5 Torus
9.6 Polynomial Curves
9.6.1 Bezier Curves
9.6.2 B-Spline Curves
9.6.3 NURBS Curves
9.7 Polynomial
Surfaces
9.7.1 Bezier Surfaces
9.7.2 B-Spline Surfaces
9.7.3 NURBS Surfaces
Chapter 10 Distance in 3D
10.1
Introduction
10.2 Point to Linear Component
10.2.1 Point to Ray or Line Segment
10.2.2 Point to Polyline
10.3 Point
to Planar Component
10.3.1 Point to Plane
10.3.2 Point to Triangle
10.3.3 Point to Rectangle
10.3.4 Point
to Polygon
10.3.5 Point to Circle or Disk
10.4 Point to Polyhedron
10.4.1 General Problem
10.4.2 Point to Oriented
Bounding Box
10.4.3 Point to Orthogonal Frustum
10.5 Point to Quadric Surface
10.5.1 Point to General Quadric Surface
10.5.2 Point to Ellipsoid
10.6 Point to Polynomial Curve
10.7 Point to Polynomial Surface
10.8 Linear Components
10.8.1 Lines and Lines
10.8.2 Segment/Segment, Line/Ray, Line/Segment, Ray/Ray, Ray/Segment
10.8.3 Segment to Segment,
Alternative Approach
10.9 Linear Component to Triangle, Rectangle, Tetrahedron, Oriented Box
10.9.1 Linear Component to Triangle
10.9.2 Linear Component to Rectangle
10.9.3 Linear Component to Tetrahedron
10.9.4 Linear Component to Oriented
Bounding Box
10.10 Line to Quadric Surface
10.11 Line to Polynomial Surface
10.12 GJK Algorithm
10.13 Miscellaneous
10.13.1 Distance between Line and Planar Curve
10.13.2 Distance between Line and Planar Solid Object
10.13.3 Distance
between Planar Curves
10.13.4 Geodesic Distance on Surfaces
Chapter 11 Intersection in 3D
11.1 Linear Components and Planar
Components
11.1.1 Linear Components and Planes
11.1.2 Linear Components and Triangles
11.1.3 Linear Components and
Polygons
11.1.4 Linear Component and Disk
11.2 Linear Components and Polyhedra
11.3 Linear Components and Quadric Surfaces
11.3.1 General Quadric Surfaces
11.3.2 Linear Components and a Sphere
11.3.3 Linear Components and an Ellipsoid
11.3.4 Linear Components and Cylinders
11.3.5 Linear Components and a Cone
11.4 Linear Components and Polynomial Surfaces
11.4.1 Algebraic Surfaces
11.4.2 Free-Form Surfaces
11.5 Planar Components
11.5.1 Two Planes
11.5.2 Three
Planes
11.5.3 Triangle and Plane
11.5.4 Triangle and Triangle
11.6 Planar Components and Polyhedra
11.6.1 Trimeshes
11.6.2 General Polyhedra
11.7 Planar Components and Quadric Surface
11.7.1 Plane and General Quadric Surface
11.7.2 Plane and Sphere
11.7.3 Plane and Cylinder
11.7.4 Plane and Cone
11.7.5 Triangle and Cone
11.8 Planar
Components and Polynomial Surfaces
11.8.1 Hermite Curves
11.8.2 Geometry Definitions
11.8.3 Computing the Curves
11.8.4 The Algorithm
11.8.5 Implementation Notes
11.9 Quadric Surfaces
11.9.1 General Intersection
11.9.2
Ellipsoids
11.10 Polynomial Surfaces
11.10.1 Subdivision Methods
11.10.2 Lattice Evaluation
11.10.3 Analytic
Methods
11.10.4 Marching Methods
11.11 The Method of Separating Axes
11.11.1 Separation of Stationary Convex Polyhedra
11.11.2 Separation of Moving Convex Polyhedra
11.11.3 Intersection Set for Stationary Convex Polyhedra
11.11.4 Contact
Set for Moving Convex Polyhedra
11.12 Miscellaneous
11.12.1 Oriented Bounding Box and Orthogonal Frustum
11.12.2 Linear
Component and Axis-Aligned Bounding Box
11.12.3 Linear Component and Oriented Bounding Box
11.12.4 Plane and Axis-Aligned
Bounding Box
11.12.5 Plane and Oriented Bounding Box
11.12.6 Axis-Aligned Bounding Boxes
11.12.7 Oriented Bounding
Boxes
11.12.8 Sphere and Axis-Aligned Bounding Box
11.12.9 Cylinders
11.12.10 Linear Component and Torus
Chapter
12 Miscellaneous 3D Problems
12.1 Projection of a Point onto a Plane
12.2 Projection of a Vector onto a Plane
12.3 Angle
between a Line and a Plane
12.4 Angle between Two Planes
12.5 Plane Normal to a Line and through a Given Point
12.6 Plane
through Three Points
12.7 Angle between Two Lines
Chapter 13 Computational Geometry Topics
13.1 Binary Space-Partitioning Trees
in 2D
13.1.1 BSP Tree Representation of a Polygon
13.1.2 Minimum Splits versus Balanced Trees
13.1.3 Point in Polygon
Using BSP Trees
13.1.4 Partitioning a Line Segment by a BSP Tree
13.2 Binary Space-Partitioning Trees in 3D
13.2.1
BSP Tree Representation of a Polyhedron
13.2.2 Minimum Splits versus Balanced Trees
13.2.3 Point in Polyhedron Using BSP
Trees
13.2.4 Partitioning a Line Segment by a BSP Tree
13.2.5 Partitioning a Convex Polygon by a BSP Tree
13.3 Point
in Polygon
13.3.1 Point in Triangle
13.3.2 Point in Convex Polygon
13.3.3 Point in General Polygon
13.3.4
Faster Point in General Polygon
13.3.5 A Grid Method
13.4 Point in Polyhedron
13.4.1 Point in Tetrahedron
13.4.2
Point in Convex Polyhedron
13.4.3 Point in General Polyhedron
13.5 Boolean Operations on Polygons
13.5.1 The Abstract
Operations
13.5.2 The Two Primitive Operations
13.5.3 Boolean Operations Using BSP Trees
13.5.4 Other Algorithms
13.6 Boolean Operations on Polyhedra
13.6.1 Abstract Operations
13.6.2 Boolean Operations Using BSP Trees
13.7
Convex Hulls
13.7.1 Convex Hulls in 2D
13.7.2 Convex Hulls in 3D
13.7.3 Convex Hulls in Higher Dimensions
13.8
Delaunay Triangulation
13.8.1 Incremental Construction in 2D
13.8.2 Incremental Construction in General Dimensions
13.8.3 Construction by Convex Hull
13.9 Polygon Partitioning
13.9.1 Visibility Graph of a Simple Polygon
13.9.2
Triangulation
13.9.3 Triangulation by Horizontal Decomposition
13.9.4 Convex Partitioning
13.10 Circumscribed and Inscribed
Balls
13.10.1 Circumscribed Ball
13.10.2 Inscribed Ball
13.11 Minimum Bounds for Point Set
13.11.1 Minimum-Area
Rectangle
13.11.2 Minimum-Volume Box
13.11.3 Minimum-Area Circle
13.11.4 Minimum-Volume Sphere
13.11.5 Miscellaneous
13.12 Area and Volume Measurements
13.12.1 Area of a 2D Polygon
13.12.2 Area of a 3D Polygon
13.12.3 Volume
of a Polyhedron
Appendix A Numerical Methods
A.1 Solving Linear Systems
A.1.1 Special Case: Solving a Triangular System
A.1.2
Gaussian Elimination
A.2 Systems of Polynomials
A.2.1 Linear Equations in One Formal Variable
A.2.2 Any-Degree Equations in
One Formal Variable
A.2.3 Any-Degree Equations in Any Formal Variables
A.3 Matrix Decompositions
A.3.1 Euler Angle Factorization
A.3.2 QR Decomposition
A.3.3 Eigendecomposition
A.3.4 Polar Decomposition
A.3.5 Singular Value Decomposition
A.4 Representations
of 3D Rotations
A.4.1 Matrix Representation
A.4.2 Axis-Angle Representation
A.4.3 Quaternion Representation
A.4.4 Performance
Issues
A.5 Root Finding
A.5.1 Methods in One Dimension
A.5.2 Methods in Many Dimensions
A.5.3 Stable Solution to Quadratic
Equations
A.6 Minimization
A.6.1 Methods in One Dimension
A.6.2 Methods in Many Dimensions
A.6.3 Minimizing a Quadratic
Form
A.6.4 Minimizing a Restricted Quadratic Form
A.7 Least Squares Fitting
A.7.1 Linear Fitting of Points (x, f (x))
A.7.2
Linear Fitting of Points Using Orthogonal Regression
A.7.3 Planar Fitting of Points (x, y, f (x, y))
A.7.4 Hyperplanar Fitting
of Points Using Orthogonal Regression
A.7.5 Fitting a Circle to 2D Points
A.7.6 Fitting a Sphere to 3D Points
A.7.7 Fitting
a Quadratic Curve to 2D Points
A.7.8 Fitting a Quadric Surface to 3D Points
A.8 Subdivision of Curves
A.8.1 Subdivision by Uniform
Sampling
A.8.2 Subdivision by Arc Length
A.8.3 Subdivision by Midpoint Distance
A.8.4 Subdivision by Variation
A.9 Topics
from Calculus
A.9.1 Level Sets
A.9.2 Minima and Maxima of Functions
A.9.3 Lagrange Multipliers
Appendix B Trigonometry
B.1 Introduction
B.1.1 Terminology
B.1.2 Angles
B.1.3 Conversion Examples
B.2 Trigonometric Functions
B.2.1 Definitions
in Terms of Exponentials
B.2.2 Domains and Ranges
B.2.3 Graphs of Trigonometric Functions
B.2.4 Derivatives of Trigonometric
Functions
B.2.5 Integration
B.3 Trigonometric Identities and Laws
B.3.1 Periodicity
B.3.2 Laws
B.3.3 Formulas
B.4 Inverse
Trigonometric Functions
B.4.1 Defining arcsin and arccos in Terms of arctan
B.4.2 Domains and Ranges
B.4.3 Graphs
B.4.4
Derivatives
B.4.5 Integration
B.5 Further Reading
Appendix C Basic Formulas for Geometric Primitives
C.1 Introduction
C.2 Triangles
C.2.1 Symbols
C.2.2 Definitions
C.2.3 Right Triangles
C.2.4 Equilateral Triangle
C.2.5 General Triangle
C.3 Quadrilaterals
C.3.1 Square
C.3.2 Rectangle
C.3.3 Parallelogram
C.3.4 Rhombus
C.3.5 Trapezoid
C.3.6 General Quadrilateral
C.4
Circles
C.4.1 Symbols
C.4.2 Full Circle
C.4.3 Sector of a Circle
C.4.4 Segment of a Circle
C.5 Polyhedra
C.5.1 Symbols
C.5.2 Box
C.5.3 Prism
C.5.4 Pyramid
C.6 Cylinder
C.7 Cone
C.8 Spheres
C.8.1 Segments
C.8.2 Sector
C.9 Torus
References
Index
About the Authors
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