Geometric Tools for Computer Graphics - 1st Edition - ISBN: 9781558605947, 9780080478029

Geometric Tools for Computer Graphics

1st Edition

Authors: Philip Schneider Philip Schneider David Eberly
eBook ISBN: 9780080478029
Hardcover ISBN: 9781558605947
Imprint: Morgan Kaufmann
Published Date: 26th September 2002
Page Count: 1056
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Table of Contents

Foreword<BR id=""CRLF"">Figures<BR id=""CRLF"">Tables<BR id=""CRLF"">Preface <BR id=""CRLF""><BR id=""CRLF"">Chapter 1 Introduction <BR id=""CRLF""> 1.1 How to Use This Book <BR id=""CRLF""> 1.2 Issues of Numerical Computation <BR id=""CRLF""> 1.2.1 Low-Level Issues <BR id=""CRLF""> 1.2.2 High-Level Issues <BR id=""CRLF""> 1.3 A Summary of the Chapters <BR id=""CRLF""><BR id=""CRLF"">Chapter 2 Matrices and Linear Systems <BR id=""CRLF""> 2.1 Introduction <BR id=""CRLF""> 2.1.1 Motivation <BR id=""CRLF""> 2.1.2 Organization <BR id=""CRLF""> 2.1.3 Notational Conventions <BR id=""CRLF""> 2.2 Tuples <BR id=""CRLF""> 2.2.1 Definition <BR id=""CRLF""> 2.2.2 Arithmetic Operations <BR id=""CRLF""> 2.3 Matrices <BR id=""CRLF""> 2.3.1 Notation and Terminology <BR id=""CRLF""> 2.3.2 Transposition <BR id=""CRLF""> 2.3.3 Arithmetic Operations <BR id=""CRLF""> 2.3.4 Matrix Multiplication <BR id=""CRLF""> 2.4 Linear Systems <BR id=""CRLF""> 2.4.1 Linear Equations <BR id=""CRLF""> 2.4.2 Linear Systems in Two Unknowns <BR id=""CRLF""> 2.4.3 General Linear Systems <BR id=""CRLF""> 2.4.4 Row Reductions, Echelon Form, and Rank <BR id=""CRLF""> 2.5 Square Matrices <BR id=""CRLF""> 2.5.1 Diagonal Matrices <BR id=""CRLF""> 2.5.2 Triangular Matrices <BR id=""CRLF""> 2.5.3 The Determinant <BR id=""CRLF""> 2.5.4 Inverse <BR id=""CRLF""> 2.6 Linear Spaces <BR id=""CRLF""> 2.6.1 Fields <BR id=""CRLF""> 2.6.2 Definition and Properties <BR id=""CRLF""> 2.6.3 Subspaces <BR id=""CRLF""> 2.6.4 Linear Combinations and Span <BR id=""CRLF""> 2.6.5 Linear Independence, Dimension, and Basis <BR id=""CRLF""> 2.7 Linear Mappings <BR id=""CRLF""> 2.7.1 Mappings in General <BR id=""CRLF""> 2.7.2 Linear Mappings <BR id=""CRLF""> 2.7.3 Matrix Representation of Linear Mappings <BR id=""CRLF""> 2.7.4 Cramer’s Rule <BR id=""CRLF""> 2.8 Eigenvalues and Eigenvectors <BR id=""CRLF""> 2.9 Euclidean Space <BR id=""CRLF""> 2.9.1 Inner Product Spaces <BR id=""CRLF""> 2.9.2 Orthogonality and Orthonormal Sets <BR id=""CRLF""> 2.10 Least Squares <BR id=""CRLF""> Recommended Reading <BR id=""CRLF""><BR id=""CRLF"">Chapter 3 Vector Algebra <BR id=""CRLF""> 3.1 Vector Basics <BR id=""CRLF""> 3.1.1 Vector Equivalence <BR id=""CRLF""> 3.1.2 Vector Addition <BR id=""CRLF""> 3.1.3 Vector Subtraction <BR id=""CRLF""> 3.1.4 Vector Scaling <BR id=""CRLF""> 3.1.5 Properties of Vector Addition and Scalar Multiplication <BR id=""CRLF""> 3.2 Vector Space <BR id=""CRLF""> 3.2.1 Span <BR id=""CRLF""> 3.2.2 Linear Independence <BR id=""CRLF""> 3.2.3 Basis, Subspaces, and Dimension <BR id=""CRLF""> 3.2.4 Orientation <BR id=""CRLF""> 3.2.5 Change of Basis <BR id=""CRLF""> 3.2.6 Linear Transformations <BR id=""CRLF""> 3.3 Affine Spaces <BR id=""CRLF""> 3.3.1 Euclidean Geometry <BR id=""CRLF""> 3.3.2 Volume, the Determinant, and the Scalar Triple Product <BR id=""CRLF""> 3.3.3 Frames <BR id=""CRLF""> 3.4 Affine Transformations <BR id=""CRLF""> 3.4.1 Types of Affine Maps <BR id=""CRLF""> 3.4.2 Composition of Affine Maps <BR id=""CRLF""> 3.5 Barycentric Coordinates and Simplexes <BR id=""CRLF""> 3.5.1 Barycentric Coordinates and Subspaces <BR id=""CRLF""> 3.5.2 Affine Independence <BR id=""CRLF""><BR id=""CRLF"">Chapter 4 Matrices, Vector Algebra, and Transformations <BR id=""CRLF""> 4.1 Introduction <BR id=""CRLF""> 4.2 Matrix Representation of Points and Vectors <BR id=""CRLF""> 4.3 Addition, Subtraction, and Multiplication <BR id=""CRLF""> 4.3.1 Vector Addition and Subtraction <BR id=""CRLF""> 4.3.2 Point and Vector Addition and Subtraction <BR id=""CRLF""> 4.3.3 Subtraction of Points <BR id=""CRLF""> 4.3.4 Scalar Multiplication <BR id=""CRLF""> 4.4 Products of Vectors <BR id=""CRLF""> 4.4.1 Dot Product <BR id=""CRLF""> 4.4.2 Cross Product <BR id=""CRLF""> 4.4.3 Tensor Product <BR id=""CRLF""> 4.4.4 The “Perp” Operator and the “Perp” Dot Product <BR id=""CRLF""> 4.5 Matrix Representation of Affine Transformations <BR id=""CRLF""> 4.6 Change-of-Basis/Frame/Coordinate System <BR id=""CRLF""> 4.7 Vector Geometry of Affine Transformations <BR id=""CRLF""> 4.7.1 Notation <BR id=""CRLF""> 4.7.2 Translation <BR id=""CRLF""> 4.7.3 Rotation <BR id=""CRLF""> 4.7.4 Scaling <BR id=""CRLF""> 4.7.5 Reflection <BR id=""CRLF""> 4.7.6 Shearing <BR id=""CRLF""> 4.8 Projections <BR id=""CRLF""> 4.8.1 Orthographic <BR id=""CRLF""> 4.8.2 Oblique <BR id=""CRLF""> 4.8.3 Perspective <BR id=""CRLF""> 4.9 Transforming Normal Vectors <BR id=""CRLF""> Recommended Reading <BR id=""CRLF""><BR id=""CRLF"">Chapter 5 Geometric Primitives in 2D <BR id=""CRLF""> 5.1 Linear Components <BR id=""CRLF""> 5.1.1 Implicit Form <BR id=""CRLF""> 5.1.2 Parametric Form <BR id=""CRLF""> 5.1.3 Converting between Representations <BR id=""CRLF""> 5.2 Triangles <BR id=""CRLF""> 5.3 Rectangles <BR id=""CRLF""> 5.4 Polylines and Polygons <BR id=""CRLF""> 5.5 Quadratic Curves <BR id=""CRLF""> 5.5.1 Circles <BR id=""CRLF""> 5.5.2 Ellipses <BR id=""CRLF""> 5.6 Polynomial Curves <BR id=""CRLF""> 5.6.1 B´ezier Curves <BR id=""CRLF""> 5.6.2 B-Spline Curves <BR id=""CRLF""> 5.6.3 NURBS Curves <BR id=""CRLF""><BR id=""CRLF"">Chapter 6 Distance in 2D <BR id=""CRLF""> 6.1 Point to Linear Component <BR id=""CRLF""> 6.1.1 Point to Line <BR id=""CRLF""> 6.1.2 Point to Ray <BR id=""CRLF""> 6.1.3 Point to Segment <BR id=""CRLF""> 6.2 Point to Polyline <BR id=""CRLF""> 6.3 Point to Polygon <BR id=""CRLF""> 6.3.1 Point to Triangle <BR id=""CRLF""> 6.3.2 Point to Rectangle <BR id=""CRLF""> 6.3.3 Point to Orthogonal Frustum <BR id=""CRLF""> 6.3.4 Point to Convex Polygon <BR id=""CRLF""> 6.4 Point to Quadratic Curve <BR id=""CRLF""> 6.5 Point to Polynomial Curve <BR id=""CRLF""> 6.6 Linear Components <BR id=""CRLF""> 6.6.1 Line to Line <BR id=""CRLF""> 6.6.2 Line to Ray <BR id=""CRLF""> 6.6.3 Line to Segment <BR id=""CRLF""> 6.6.4 Ray to Ray <BR id=""CRLF""> 6.6.5 Ray to Segment <BR id=""CRLF""> 6.6.6 Segment to Segment <BR id=""CRLF""> 6.7 Linear Component to Polyline or Polygon <BR id=""CRLF""> 6.8 Linear Component to Quadratic Curve <BR id=""CRLF""> 6.9 Linear Component to Polynomial Curve <BR id=""CRLF""> 6.10 GJK Algorithm <BR id=""CRLF""> 6.10.1 Set Operations <BR id=""CRLF""> 6.10.2 Overview of the Algorithm <BR id=""CRLF""> 6.10.3 Alternatives to GJK <BR id=""CRLF""><BR id=""CRLF"">Chapter 7 Intersection in 2D <BR id=""CRLF""> 7.1 Linear Components <BR id=""CRLF""> 7.2 Linear Components and Polylines <BR id=""CRLF""> 7.3 Linear Components and Quadratic Curves <BR id=""CRLF""> 7.3.1 Linear Components and General Quadratic Curves <BR id=""CRLF""> 7.3.2 Linear Components and Circular Components <BR id=""CRLF""> 7.4 Linear Components and Polynomial Curves <BR id=""CRLF""> 7.4.1 Algebraic Method <BR id=""CRLF""> 7.4.2 Polyline Approximation <BR id=""CRLF""> 7.4.3 Hierarchical Bounding <BR id=""CRLF""> 7.4.4 Monotone Decomposition <BR id=""CRLF""> 7.4.5 Rasterization<BR id=""CRLF""> 7.5 Quadratic Curves <BR id=""CRLF""> 7.5.1 General Quadratic Curves <BR id=""CRLF""> 7.5.2 Circular Components <BR id=""CRLF""> 7.5.3 Ellipses <BR id=""CRLF""> 7.6 Polynomial Curves <BR id=""CRLF""> 7.6.1 Algebraic Method <BR id=""CRLF""> 7.6.2 Polyline Approximation<BR id=""CRLF""> 7.6.3 Hierarchical Bounding <BR id=""CRLF""> 7.6.4 Rasterization <BR id=""CRLF""> 7.7 The Method of Separating Axes <BR id=""CRLF""> 7.7.1 Separation by Projection onto a Line <BR id=""CRLF""> 7.7.2 Separation of Stationary Convex Polygons <BR id=""CRLF""> 7.7.3 Separation of Moving Convex Polygons <BR id=""CRLF""> 7.7.4 Intersection Set for Stationary Convex Polygons <BR id=""CRLF""> 7.7.5 Contact Set for Moving Convex Polygons <BR id=""CRLF""><BR id=""CRLF"">Chapter 8 Miscellaneous 2D Problems <BR id=""CRLF""> 8.1 Circle through Three Points <BR id=""CRLF""> 8.2 Circle Tangent to Three Lines <BR id=""CRLF""> 8.3 Line Tangent to a Circle at a Given Point <BR id=""CRLF""> 8.4 Line Tangent to a Circle through a Given Point <BR id=""CRLF""> 8.5 Lines Tangent to Two Circles <BR id=""CRLF""> 8.6 Circle through Two Points with a Given Radius <BR id=""CRLF""> 8.7 Circle through a Point and Tangent to a Line with a Given Radius <BR id=""CRLF""> 8.8 Circles Tangent to Two Lines with a Given Radius <BR id=""CRLF""> 8.9 Circles through a Point and Tangent to a Circle with a Given Radius <BR id=""CRLF""> 8.10 Circles Tangent to a Line and a Circle with a Given Radius <BR id=""CRLF""> 8.11 Circles Tangent to Two Circles with a Given Radius <BR id=""CRLF""> 8.12 Line Perpendicular to a Given Line through a Given Point <BR id=""CRLF""> 8.13 Line between and Equidistant to Two Points <BR id=""CRLF""> 8.14 Line Parallel to a Given Line at a Given Distance <BR id=""CRLF""> 8.15 Line Parallel to a Given Line at a Given Vertical (Horizontal) Distance <BR id=""CRLF""> 8.16 Lines Tangent to a Given Circle and Normal to a Given Line <BR id=""CRLF""><BR id=""CRLF"">Chapter 9 Geometric Primitives in 3D <BR id=""CRLF""> 9.1 Linear Components <BR id=""CRLF""> 9.2 Planar Components <BR id=""CRLF""> 9.2.1 Planes <BR id=""CRLF""> 9.2.2 Coordinate System Relative to a Plane <BR id=""CRLF""> 9.2.3 2D Objects in a Plane <BR id=""CRLF""> 9.3 Polymeshes, Polyhedra, and Polytopes <BR id=""CRLF""> 9.3.1 Vertex-Edge-Face Tables <BR id=""CRLF""> 9.3.2 Connected Meshes <BR id=""CRLF""> 9.3.3 Manifold Meshes <BR id=""CRLF""> 9.3.4 Closed Meshes <BR id=""CRLF""> 9.3.5 Consistent Ordering <BR id=""CRLF""> 9.3.6 Platonic Solids <BR id=""CRLF""> 9.4 Quadric Surfaces <BR id=""CRLF""> 9.4.1 Three Nonzero Eigenvalues <BR id=""CRLF""> 9.4.2 Two Nonzero Eigenvalues <BR id=""CRLF""> 9.4.3 One Nonzero Eigenvalue <BR id=""CRLF""> 9.5 Torus <BR id=""CRLF""> 9.6 Polynomial Curves <BR id=""CRLF""> 9.6.1 Bézier Curves <BR id=""CRLF""> 9.6.2 B-Spline Curves <BR id=""CRLF""> 9.6.3 NURBS Curves <BR id=""CRLF""> 9.7 Polynomial Surfaces <BR id=""CRLF""> 9.7.1 Bézier Surfaces <BR id=""CRLF""> 9.7.2 B-Spline Surfaces <BR id=""CRLF""> 9.7.3 NURBS Surfaces <BR id=""CRLF""><BR id=""CRLF"">Chapter 10 Distance in 3D <BR id=""CRLF""> 10.1 Introduction <BR id=""CRLF""> 10.2 Point to Linear Component <BR id=""CRLF""> 10.2.1 Point to Ray or Line Segment <BR id=""CRLF""> 10.2.2 Point to Polyline <BR id=""CRLF""> 10.3 Point to Planar Component <BR id=""CRLF""> 10.3.1 Point to Plane <BR id=""CRLF""> 10.3.2 Point to Triangle <BR id=""CRLF""> 10.3.3 Point to Rectangle <BR id=""CRLF""> 10.3.4 Point to Polygon <BR id=""CRLF""> 10.3.5 Point to Circle or Disk <BR id=""CRLF""> 10.4 Point to Polyhedron <BR id=""CRLF""> 10.4.1 General Problem <BR id=""CRLF""> 10.4.2 Point to Oriented Bounding Box <BR id=""CRLF""> 10.4.3 Point to Orthogonal Frustum <BR id=""CRLF""> 10.5 Point to Quadric Surface <BR id=""CRLF""> 10.5.1 Point to General Quadric Surface <BR id=""CRLF""> 10.5.2 Point to Ellipsoid <BR id=""CRLF""> 10.6 Point to Polynomial Curve <BR id=""CRLF""> 10.7 Point to Polynomial Surface <BR id=""CRLF""> 10.8 Linear Components <BR id=""CRLF""> 10.8.1 Lines and Lines <BR id=""CRLF""> 10.8.2 Segment/Segment, Line/Ray, Line/Segment, Ray/Ray, Ray/Segment <BR id=""CRLF""> 10.8.3 Segment to Segment, Alternative Approach <BR id=""CRLF""> 10.9 Linear Component to Triangle, Rectangle, Tetrahedron, Oriented Box <BR id=""CRLF""> 10.9.1 Linear Component to Triangle <BR id=""CRLF""> 10.9.2 Linear Component to Rectangle <BR id=""CRLF""> 10.9.3 Linear Component to Tetrahedron <BR id=""CRLF""> 10.9.4 Linear Component to Oriented Bounding Box <BR id=""CRLF""> 10.10 Line to Quadric Surface <BR id=""CRLF""> 10.11 Line to Polynomial Surface <BR id=""CRLF""> 10.12 GJK Algorithm <BR id=""CRLF""> 10.13 Miscellaneous <BR id=""CRLF""> 10.13.1 Distance between Line and Planar Curve <BR id=""CRLF""> 10.13.2 Distance between Line and Planar Solid Object <BR id=""CRLF""> 10.13.3 Distance between Planar Curves <BR id=""CRLF""> 10.13.4 Geodesic Distance on Surfaces <BR id=""CRLF""><BR id=""CRLF"">Chapter 11 Intersection in 3D <BR id=""CRLF""> 11.1 Linear Components and Planar Components <BR id=""CRLF""> 11.1.1 Linear Components and Planes <BR id=""CRLF""> 11.1.2 Linear Components and Triangles <BR id=""CRLF""> 11.1.3 Linear Components and Polygons <BR id=""CRLF""> 11.1.4 Linear Component and Disk <BR id=""CRLF""> 11.2 Linear Components and Polyhedra <BR id=""CRLF""> 11.3 Linear Components and Quadric Surfaces <BR id=""CRLF""> 11.3.1 General Quadric Surfaces <BR id=""CRLF""> 11.3.2 Linear Components and a Sphere <BR id=""CRLF""> 11.3.3 Linear Components and an Ellipsoid <BR id=""CRLF""> 11.3.4 Linear Components and Cylinders <BR id=""CRLF""> 11.3.5 Linear Components and a Cone <BR id=""CRLF""> 11.4 Linear Components and Polynomial Surfaces <BR id=""CRLF""> 11.4.1 Algebraic Surfaces <BR id=""CRLF""> 11.4.2 Free-Form Surfaces <BR id=""CRLF""> 11.5 Planar Components <BR id=""CRLF""> 11.5.1 Two Planes <BR id=""CRLF""> 11.5.2 Three Planes <BR id=""CRLF""> 11.5.3 Triangle and Plane <BR id=""CRLF""> 11.5.4 Triangle and Triangle <BR id=""CRLF""> 11.6 Planar Components and Polyhedra <BR id=""CRLF""> 11.6.1 Trimeshes <BR id=""CRLF""> 11.6.2 General Polyhedra <BR id=""CRLF""> 11.7 Planar Components and Quadric Surface<BR id=""CRLF""> 11.7.1 Plane and General Quadric Surface <BR id=""CRLF""> 11.7.2 Plane and Sphere <BR id=""CRLF""> 11.7.3 Plane and Cylinder <BR id=""CRLF""> 11.7.4 Plane and Cone <BR id=""CRLF""> 11.7.5 Triangle and Cone <BR id=""CRLF""> 11.8 Planar Components and Polynomial Surfaces <BR id=""CRLF""> 11.8.1 Hermite Curves <BR id=""CRLF""> 11.8.2 Geometry Definitions <BR id=""CRLF""> 11.8.3 Computing the Curves <BR id=""CRLF""> 11.8.4 The Algorithm <BR id=""CRLF""> 11.8.5 Implementation Notes <BR id=""CRLF""> 11.9 Quadric Surfaces <BR id=""CRLF""> 11.9.1 General Intersection <BR id=""CRLF""> 11.9.2 Ellipsoids <BR id=""CRLF""> 11.10 Polynomial Surfaces <BR id=""CRLF""> 11.10.1 Subdivision Methods <BR id=""CRLF""> 11.10.2 Lattice Evaluation <BR id=""CRLF""> 11.10.3 Analytic Methods <BR id=""CRLF""> 11.10.4 Marching Methods <BR id=""CRLF""> 11.11 The Method of Separating Axes <BR id=""CRLF""> 11.11.1 Separation of Stationary Convex Polyhedra <BR id=""CRLF""> 11.11.2 Separation of Moving Convex Polyhedra <BR id=""CRLF""> 11.11.3 Intersection Set for Stationary Convex Polyhedra <BR id=""CRLF""> 11.11.4 Contact Set for Moving Convex Polyhedra <BR id=""CRLF""> 11.12 Miscellaneous <BR id=""CRLF""> 11.12.1 Oriented Bounding Box and Orthogonal Frustum <BR id=""CRLF""> 11.12.2 Linear Component and Axis-Aligned Bounding Box <BR id=""CRLF""> 11.12.3 Linear Component and Oriented Bounding Box <BR id=""CRLF""> 11.12.4 Plane and Axis-Aligned Bounding Box <BR id=""CRLF""> 11.12.5 Plane and Oriented Bounding Box <BR id=""CRLF""> 11.12.6 Axis-Aligned Bounding Boxes <BR id=""CRLF""> 11.12.7 Oriented Bounding Boxes <BR id=""CRLF""> 11.12.8 Sphere and Axis-Aligned Bounding Box <BR id=""CRLF""> 11.12.9 Cylinders <BR id=""CRLF""> 11.12.10 Linear Component and Torus <BR id=""CRLF""><BR id=""CRLF"">Chapter 12 Miscellaneous 3D Problems <BR id=""CRLF""> 12.1 Projection of a Point onto a Plane <BR id=""CRLF""> 12.2 Projection of a Vector onto a Plane <BR id=""CRLF""> 12.3 Angle between a Line and a Plane <BR id=""CRLF""> 12.4 Angle between Two Planes <BR id=""CRLF""> 12.5 Plane Normal to a Line and through a Given Point <BR id=""CRLF""> 12.6 Plane through Three Points <BR id=""CRLF""> 12.7 Angle between Two Lines <BR id=""CRLF""><BR id=""CRLF"">Chapter 13 Computational Geometry Topics <BR id=""CRLF""> 13.1 Binary Space-Partitioning Trees in 2D <BR id=""CRLF""> 13.1.1 BSP Tree Representation of a Polygon <BR id=""CRLF""> 13.1.2 Minimum Splits versus Balanced Trees <BR id=""CRLF""> 13.1.3 Point in Polygon Using BSP Trees <BR id=""CRLF""> 13.1.4 Partitioning a Line Segment by a BSP Tree <BR id=""CRLF""> 13.2 Binary Space-Partitioning Trees in 3D <BR id=""CRLF""> 13.2.1 BSP Tree Representation of a Polyhedron <BR id=""CRLF""> 13.2.2 Minimum Splits versus Balanced Trees <BR id=""CRLF""> 13.2.3 Point in Polyhedron Using BSP Trees <BR id=""CRLF""> 13.2.4 Partitioning a Line Segment by a BSP Tree <BR id=""CRLF""> 13.2.5 Partitioning a Convex Polygon by a BSP Tree <BR id=""CRLF""> 13.3 Point in Polygon <BR id=""CRLF""> 13.3.1 Point in Triangle <BR id=""CRLF""> 13.3.2 Point in Convex Polygon <BR id=""CRLF""> 13.3.3 Point in General Polygon <BR id=""CRLF""> 13.3.4 Faster Point in General Polygon <BR id=""CRLF""> 13.3.5 A Grid Method <BR id=""CRLF""> 13.4 Point in Polyhedron <BR id=""CRLF""> 13.4.1 Point in Tetrahedron <BR id=""CRLF""> 13.4.2 Point in Convex Polyhedron <BR id=""CRLF""> 13.4.3 Point in General Polyhedron <BR id=""CRLF""> 13.5 Boolean Operations on Polygons <BR id=""CRLF""> 13.5.1 The Abstract Operations <BR id=""CRLF""> 13.5.2 The Two Primitive Operations <BR id=""CRLF""> 13.5.3 Boolean Operations Using BSP Trees <BR id=""CRLF""> 13.5.4 Other Algorithms <BR id=""CRLF""> 13.6 Boolean Operations on Polyhedra <BR id=""CRLF""> 13.6.1 Abstract Operations <BR id=""CRLF""> 13.6.2 Boolean Operations Using BSP Trees <BR id=""CRLF""> 13.7 Convex Hulls <BR id=""CRLF""> 13.7.1 Convex Hulls in 2D <BR id=""CRLF""> 13.7.2 Convex Hulls in 3D <BR id=""CRLF""> 13.7.3 Convex Hulls in Higher Dimensions <BR id=""CRLF""> 13.8 Delaunay Triangulation <BR id=""CRLF""> 13.8.1 Incremental Construction in 2D <BR id=""CRLF""> 13.8.2 Incremental Construction in General Dimensions <BR id=""CRLF""> 13.8.3 Construction by Convex Hull <BR id=""CRLF""> 13.9 Polygon Partitioning <BR id=""CRLF""> 13.9.1 Visibility Graph of a Simple Polygon <BR id=""CRLF""> 13.9.2 Triangulation <BR id=""CRLF""> 13.9.3 Triangulation by Horizontal Decomposition <BR id=""CRLF""> 13.9.4 Convex Partitioning <BR id=""CRLF""> 13.10 Circumscribed and Inscribed Balls <BR id=""CRLF""> 13.10.1 Circumscribed Ball <BR id=""CRLF""> 13.10.2 Inscribed Ball <BR id=""CRLF""> 13.11 Minimum Bounds for Point Set<BR id=""CRLF""> 13.11.1 Minimum-Area Rectangle <BR id=""CRLF""> 13.11.2 Minimum-Volume Box <BR id=""CRLF""> 13.11.3 Minimum-Area Circle <BR id=""CRLF""> 13.11.4 Minimum-Volume Sphere <BR id=""CRLF""> 13.11.5 Miscellaneous <BR id=""CRLF""> 13.12 Area and Volume Measurements <BR id=""CRLF""> 13.12.1 Area of a 2D Polygon <BR id=""CRLF""> 13.12.2 Area of a 3D Polygon <BR id=""CRLF""> 13.12.3 Volume of a Polyhedron <BR id=""CRLF""><BR id=""CRLF"">Appendix A Numerical Methods <BR id=""CRLF"">A.1 Solving Linear Systems <BR id=""CRLF""> A.1.1 Special Case: Solving a Triangular System <BR id=""CRLF""> A.1.2 Gaussian Elimination <BR id=""CRLF"">A.2 Systems of Polynomials <BR id=""CRLF""> A.2.1 Linear Equations in One Formal Variable <BR id=""CRLF""> A.2.2 Any-Degree Equations in One Formal Variable <BR id=""CRLF""> A.2.3 Any-Degree Equations in Any Formal Variables <BR id=""CRLF"">A.3 Matrix Decompositions <BR id=""CRLF""> A.3.1 Euler Angle Factorization <BR id=""CRLF""> A.3.2 QR Decomposition <BR id=""CRLF""> A.3.3 Eigendecomposition <BR id=""CRLF""> A.3.4 Polar Decomposition <BR id=""CRLF""> A.3.5 Singular Value Decomposition <BR id=""CRLF"">A.4 Representations of 3D Rotations <BR id=""CRLF""> A.4.1 Matrix Representation <BR id=""CRLF""> A.4.2 Axis-Angle Representation <BR id=""CRLF""> A.4.3 Quaternion Representation <BR id=""CRLF""> A.4.4 Performance Issues <BR id=""CRLF"">A.5 Root Finding <BR id=""CRLF""> A.5.1 Methods in One Dimension <BR id=""CRLF""> A.5.2 Methods in Many Dimensions <BR id=""CRLF""> A.5.3 Stable Solution to Quadratic Equations <BR id=""CRLF"">A.6 Minimization <BR id=""CRLF""> A.6.1 Methods in One Dimension <BR id=""CRLF""> A.6.2 Methods in Many Dimensions <BR id=""CRLF""> A.6.3 Minimizing a Quadratic Form <BR id=""CRLF""> A.6.4 Minimizing a Restricted Quadratic Form <BR id=""CRLF"">A.7 Least Squares Fitting <BR id=""CRLF""> A.7.1 Linear Fitting of Points (x, f (x)) <BR id=""CRLF""> A.7.2 Linear Fitting of Points Using Orthogonal Regression <BR id=""CRLF""> A.7.3 Planar Fitting of Points (x, y, f (x, y)) <BR id=""CRLF""> A.7.4 Hyperplanar Fitting of Points Using Orthogonal Regression <BR id=""CRLF""> A.7.5 Fitting a Circle to 2D Points <BR id=""CRLF""> A.7.6 Fitting a Sphere to 3D Points <BR id=""CRLF""> A.7.7 Fitting a Quadratic Curve to 2D Points <BR id=""CRLF""> A.7.8 Fitting a Quadric Surface to 3D Points <BR id=""CRLF"">A.8 Subdivision of Curves <BR id=""CRLF""> A.8.1 Subdivision by Uniform Sampling <BR id=""CRLF""> A.8.2 Subdivision by Arc Length <BR id=""CRLF""> A.8.3 Subdivision by Midpoint Distance <BR id=""CRLF""> A.8.4 Subdivision by Variation <BR id=""CRLF"">A.9 Topics from Calculus <BR id=""CRLF""> A.9.1 Level Sets <BR id=""CRLF""> A.9.2 Minima and Maxima of Functions <BR id=""CRLF""> A.9.3 Lagrange Multipliers <BR id=""CRLF""><BR id=""CRLF"">Appendix B Trigonometry <BR id=""CRLF"">B.1 Introduction <BR id=""CRLF""> B.1.1 Terminology <BR id=""CRLF""> B.1.2 Angles <BR id=""CRLF""> B.1.3 Conversion Examples <BR id=""CRLF"">B.2 Trigonometric Functions <BR id=""CRLF""> B.2.1 Definitions in Terms of Exponentials <BR id=""CRLF""> B.2.2 Domains and Ranges <BR id=""CRLF""> B.2.3 Graphs of Trigonometric Functions <BR id=""CRLF""> B.2.4 Derivatives of Trigonometric Functions <BR id=""CRLF""> B.2.5 Integration <BR id=""CRLF"">B.3 Trigonometric Identities and Laws <BR id=""CRLF""> B.3.1 Periodicity <BR id=""CRLF""> B.3.2 Laws <BR id=""CRLF""> B.3.3 Formulas <BR id=""CRLF"">B.4 Inverse Trigonometric Functions <BR id=""CRLF""> B.4.1 Defining arcsin and arccos in Terms of arctan <BR id=""CRLF""> B.4.2 Domains and Ranges <BR id=""CRLF""> B.4.3 Graphs <BR id=""CRLF""> B.4.4 Derivatives <BR id=""CRLF""> B.4.5 Integration <BR id=""CRLF"">B.5 Further Reading <BR id=""CRLF""><BR id=""CRLF"">Appendix C Basic Formulas for Geometric Primitives <BR id=""CRLF"">C.1 Introduction <BR id=""CRLF"">C.2 Triangles <BR id=""CRLF""> C.2.1 Symbols <BR id=""CRLF""> C.2.2 Definitions <BR id=""CRLF""> C.2.3 Right Triangles <BR id=""CRLF""> C.2.4 Equilateral Triangle <BR id=""CRLF""> C.2.5 General Triangle <BR id=""CRLF"">C.3 Quadrilaterals <BR id=""CRLF""> C.3.1 Square <BR id=""CRLF""> C.3.2 Rectangle <BR id=""CRLF""> C.3.3 Parallelogram <BR id=""CRLF""> C.3.4 Rhombus <BR id=""CRLF""> C.3.5 Trapezoid <BR id=""CRLF""> C.3.6 General Quadrilateral <BR id=""CRLF"">C.4 Circles <BR id=""CRLF""> C.4.1 Symbols <BR id=""CRLF""> C.4.2 Full Circle <BR id=""CRLF""> C.4.3 Sector of a Circle <BR id=""CRLF""> C.4.4 Segment of a Circle <BR id=""CRLF"">C.5 Polyhedra <BR id=""CRLF""> C.5.1 Symbols <BR id=""CRLF""> C.5.2 Box <BR id=""CRLF""> C.5.3 Prism <BR id=""CRLF""> C.5.4 Pyramid <BR id=""CRLF"">C.6 Cylinder <BR id=""CRLF"">C.7 Cone <BR id=""CRLF"">C.8 Spheres <BR id=""CRLF""> C.8.1 Segments <BR id=""CRLF""> C.8.2 Sector <BR id=""CRLF"">C.9 Torus <BR id=""CRLF""><BR id=""CRLF"">References <BR id=""CRLF"">Index <BR id=""CRLF"">About the Authors


Description

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.

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 www.mkp.com/gtcg.

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 www.mkp.com/gtcg.

Readership

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.


Details

No. of pages:
1056
Language:
English
Copyright:
© Morgan Kaufmann 2003
Published:
Imprint:
Morgan Kaufmann
eBook ISBN:
9780080478029
Hardcover ISBN:
9781558605947

Reviews

"An hour of a programmer's time often costs more than the price of a book. By this measure, you hold a volume potentially worth thousands of dollars. That it can be purchased for a fraction of this cost I consider a modern miracle. The amount of information crammed into this book is incredible." --Eric Haines

Ratings and Reviews


About the Authors

Philip Schneider Author

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

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.

Philip Schneider Author

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

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.

David Eberly Author

Dave Eberly is the 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, 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 comp.graphics.algorit

Affiliations and Expertise

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.