Geometric Tools for Computer Graphics

Geometric Tools for Computer Graphics

1st Edition - September 26, 2002

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  • Authors: Philip Schneider, David Eberly
  • Hardcover ISBN: 9781558605947
  • eBook ISBN: 9780080478029

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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.FeaturesFilled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors.Covers problems relevant for both 2D and 3D graphics programming.Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you.Provides the math and geometry background you need to understand the solutions and put them to work.Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode.Resources associated with the book are available at the companion Web site

Key Features

* Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors.
* Covers problems relevant for both 2D and 3D graphics programming.
* Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you.
* Provides the math and geometry background you need to understand the solutions and put them to work.
* Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode.
* Resources associated with the book are available at the companion Web site


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.

Table of Contents

  • Foreword

    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 Bézier Curves
    9.6.2 B-Spline Curves
    9.6.3 NURBS Curves
    9.7 Polynomial Surfaces
    9.7.1 Bézier 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

    About the Authors

Product details

  • No. of pages: 1056
  • Language: English
  • Copyright: © Morgan Kaufmann 2002
  • Published: September 26, 2002
  • Imprint: Morgan Kaufmann
  • Hardcover ISBN: 9781558605947
  • eBook ISBN: 9780080478029

About the Authors

Philip Schneider

24 years of professional programming, primarily focused on modeling tools and geometric algorithms. Employers include Digital Equipment Corporation, Apple, Walt Disney Feature Animation, Digital Domain, and Industrial Light + Magic. Formed and lead groups specializing in these areas as well as in physics simulation.

Film Credits: Oil & Vinegar, 102 Dalmatians, Disney's Magic Lamp, Mickey's Philharmagic, Reign of Fire, Kangaroo Jack, Chicken Little, Indiana Jones and the Kingdom of the Crystal Skull, Pirates of the Caribbean: Dead Man's Chest, Harry Potter and the Goblet of Fire.

ACM Siggraph, IEEE.

M.S. in Computer Science, University of Washington.

Affiliations and Expertise

Employers include Digital Equipment Corporation, Apple, Walt Disney Feature Animation, Digital Domain, and Industrial Light + Magic

David Eberly

Dave Eberly is the president of Geometric Tools, Inc. (, 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, NetImmerse. He also worked for NDL on Gamebryo, which was the next-generation engine after NetImmerse. His background includes a BA degree in mathematics from Bloomsburg University, MS and PhD degrees in mathematics from the University of Colorado at Boulder, and MS and PhD degrees in computer science from the University of North Carolina at ChapelHill. He is the author of 3D Game Engine Design, 2nd Edition (2006), 3D Game Engine Architecture (2005), Game Physics (2004), and coauthor with Philip Schneider of Geometric Tools for Computer Graphics (2003), all published by Morgan Kaufmann. As a mathematician, Dave did research in the mathematics of combustion, signal and image processing, and length-biased distributions in statistics. He was an associate professor at the University of Texas at San Antonio with an adjunct appointment in radiology at the U.T. Health Science Center at San Antonio. In 1991, he gave up his tenured position to re-train in computer science at the University of North Carolina. After graduating in 1994, he remained for one year as a research associate professor in computer science with a joint appointment in the Department of Neurosurgery, working in medical image analysis. His next stop was the SAS Institute, working for a year on SAS/Insight, a statistical graphics package. Finally, deciding that computer graphics and geometry were his real calling, Dave went to work for NDL (which is now Emergent Game Technologies), then to Magic Software, Inc., which later became Geometric Tools, Inc. Dave’s participation in the newsgroup

Affiliations and Expertise

President of Geometric Tools, Inc.

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