GEOMETRIC TOOLS FOR COMPUTER GRAPHICS

Transcription

1 GEOMETRIC TOOLS FOR COMPUTER GRAPHICS PHILIP J. SCHNEIDER DAVID H. EBERLY MORGAN KAUFMANN PUBLISHERS A N I M P R I N T O F E L S E V I E R S C I E N C E A M S T E R D A M B O S T O N L O N D O N N E W Y O R K S I N G A P O R E SYDNEY T O K Y O

2 CONTENTS FOREWORD FIGURES TABLES PREFACE Vll xxiii xli xliii 1 INTRODUCTION 1.1 How to Use This Book 1.2 Issues of Numerical Computation Low-Level Issues High-Level Issues 1.3 A Summary of the Chapters MATRICES AND LINEAR SYSTEMS Introduction Tuples Motivation Organization Notational Conventions Definition Arithmetic Operations Matrices Notation and Terminology Transposition Arithmetic Operations Matrix Multiplication Linear! Systems Linear Equations Linear Systems in Two Unknowns IX

3 x Contents General Linear Systems Row Reductions, Echelon Form, and Rank 2.5 Square Matrices Diagonal Matrices Triangular Matrices The Determinant Inverse 2.6 Linear Spaces Fields Definition and Properties Subspaces Linear Combinations and Span Linear Independence, Dimension, and Basis 2.7 Linear Mappings Mappings in General Linear Mappings Matrix Representation of Linear Mappings Cramer's Rule 2.8 Eigenvalues and Eigenvectors 2.9 Euclidean Space Inner Product Spaces Orthogonality and Orthonormal Sets 2.10 Least Squares Recommended Reading VECTOR ALGEBRA Vector Vector Basics Vector Equivalence Vector Addition Vector Subtraction Vector Scaling Properties of Vector Addition and Scalar Multiplication Space Span Linear Independence Basis, Subspaces, and Dimension Orientation Change of Basis Linear Transformations

4 Contents xi 3.3 Affine Spaces Euclidean Geometry Volume, the Determinant, and the Scalar Triple Product Frames Affine Transformations Types of Affine Maps Composition of Affine Maps Barycentric Coordinates and Simplexes Barycentric Coordinates and Subspaces Affine Independence 106 MATRICES, VECTOR ALGEBRA, AND TRANSFORMATIONS Introduction Matrix Representation of Points and Vectors Addition, Subtraction, and Multiplication Vector Addition and Subtraction Point and Vector Addition and Subtraction Subtraction of Points Scalar Multiplication Products of Vectors Matrix Dot Product Cross Product Tensor Product The "Perp" Operator and the "Perp" Dot Product Representation of Affine Transformations Change-of-Basis/Frame/Coordinate System Vector Geometry of Affine Transformations Notation Translation Rotation Scaling Reflection Shearing Projections Orthographic Oblique Perspective

5 Xll Contents 4.9 Transforming Normal Vectors Recommended Reading GEOMETRIC PRIMITIVES IN 2D 5.1 Linear Components Implicit Form Parametric Form Converting between Representations 5.2 Triangles 5.3 Rectangles 5.4 Polylines and Polygons 5.5 Quadratic Curves Circles Ellipses 5.6 Polynomial Curves Bezier Curves B-Spline Curves NURBS Curves DISTANCE IN 2D 6.1 Point to Linear Component Point to Line Point to Ray Point to Segment 6.2 Point to Polyline 6.3 Point to Polygon Point to Triangle Point to Rectangle Point to Orthogonal Frustum Point to Convex Polygon 6.4 Point to Quadratic Curve 6.5 Point to Polynomial Curve 6.6 Linear Components Line to Line Line to Ray Line to Segment

6 Contents Xlll Ray to Ray Ray to Segment 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 Set Operations Overview of the Algorithm Alternatives to GJK INTERSECTION IN 2D 7.1 Linear Components 7.2 Linear Components and Polylines 7.3 Linear Components and Quadratic Curves Linear Components and General Quadratic Curves Linear Components and Circular Components 7.4 Linear Components and Polynomial Curves Algebraic Method Polyline Approximation Hierarchical Bounding Monotone Decomposition Rasterization 7.5 Quadratic Curves General Quadratic Curves Circular Components Ellipses 7.6 Polynomial Curves Algebraic Method Polyline Approximation Hierarchical Bounding Rasterization 7.7 The Method of Separating Axes Separation by Projection onto a Line Separation of Stationary Convex Polygons Separation of Moving Convex Polygons Intersection Set for Stationary Convex Polygons Contact Set for Moving Convex Polygons

7 xiv Contents 8 MISCELLANEOUS 2D PROBLEMS Circle through Three Points Circle Tangent to Three Lines Line Tangent to a Circle at a Given Point Line Tangent to a Circle through a Given Point Lines Tangent to Two Circles Circle through Two Points with a Given Radius Circle through a Point and Tangent to a Line with a Given Radius Circles Tangent to Two Lines with a Given Radius Circles through a Point and Tangent to a Circle with a Given Radius Circles Tangent to a Line and a Circle with a Given Radius Circles Tangent to Two Circles with a Given Radius Line Perpendicular to a Given Line through a Given Point Line between and Equidistant to Two Points Line Parallel to a Given Line at a Given Distance Line Parallel to a Given Line at a Given Vertical (Horizontal) Distance Lines Tangent to a Given Circle and Normal to a Given Line 322 GEOMETRIC PRIMITIVES IN 3D Linear Components 9.2 Planar Components Planes Coordinate System Relative to a Plane D Objects in a Plane 9.3 Polymeshes, Polyhedra, and Polytopes Vertex-Edge-Face Tables Connected Meshes Manifold Meshes Closed Meshes Consistent Ordering Platonic Solids

8 Contents xv Quadric Surfaces Three Nonzero Eigenvalues Two Nonzero Eigenvalues One Nonzero Eigenvalue 9.5 Torus 9.6 Polynomial Curves Bezier Curves B-Spline Curves NURBS Curves 9.7 Polynomial Surfaces Bezier Surfaces B-Spline Surfaces NURBS Surfaces DISTANCE IN 3D 10.1 Introduction 10.2 Point to Linear Component Point to Ray or Line Segment Point to Polyline 10.3 Point to Planar Component Point to Plane Point to Triangle Point to Rectangle Point to Polygon Point to Circle or Disk 10.4 Point to Polyhedron General Problem Point to Oriented Bounding Box Point to Orthogonal Frustum 10.5 Point to Quadric Surface Point to General Quadric Surface Point to Ellipsoid 10.6 Point to Polynomial Curve 10.7 Point to Polynomial Surface 10.8 Linear Components Lines and Lines Segment/Segment, Line/Ray, Line/Segment, Ray/Ray, Ray/Segment Segment to Segment, Alternative Approach

9 XVI Contents 10.9 Linear Component to Triangle, Rectangle, Tetrahedron, Oriented Box Linear Component to Triangle Linear Component to Rectangle Linear Component to Tetrahedron Linear Component to Oriented Bounding Box Line to Quadric Surface Line to Polynomial Surface GJK Algorithm Miscellaneous Distance between Line and Planar Curve Distance between Line and Planar Solid Object Distance between Planar Curves Geodesic Distance on Surfaces INTERSECTION IN 3D Linear Components and Planar Components Linear Components and Planes Linear Components and Triangles Linear Components and Polygons Linear Component and Disk 11.2 Linear Components and Polyhedra 11.3 Linear Components and Quadric Surfaces General Quadric Surfaces Linear Components and a Sphere Linear Components and an Ellipsoid Linear Components and Cylinders Linear Components and a Cone 11.4 Linear Components and Polynomial Surfaces Algebraic Surfaces Free-Form Surfaces 11.5 Planar Components Two Planes Three Planes Triangle and Plane Triangle and Triangle 11.6 Planar Components and Polyhedra Trimeshes General Polyhedra

10 Contents xvil 11.7 Planar Components and Quadric Surfaces Plane and General Quadric Surface Plane and Sphere Plane and Cylinder Plane and Cone Triangle and Cone 11.8 Planar Components and Polynomial Surfaces Hermite Curves Geometry Definitions Computing the Curves The Algorithm Implementation Notes 11.9 Quadric Surfaces General Intersection Ellipsoids Polynomial Surfaces Subdivision Methods Lattice Evaluation Analytic Methods Marching Methods The Method of Separating Axes Separation of Stationary Convex Polyhedra Separation of Moving Convex Polyhedra Intersection Set for Stationary Convex Polyhedra Contact Set for Moving Convex Polyhedra Miscellaneous Oriented Bounding Box and Orthogonal Frustum Linear Component and Axis-Aligned Bounding Box Linear Component and Oriented Bounding Box Plane and Axis-Aligned Bounding Box Plane and Oriented Bounding Box Axis-Aligned Bounding Boxes Oriented Bounding Boxes Sphere and Axis-Aligned Bounding Box Cylinders Linear Component and Torus 12 MISCELLANEOUS 3D PROBLEMS 12.1 Projection of a Point onto a Plane 12.2 Projection of a Vector onto a Plane

11 xviii Contents 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 COMPUTATIONAL GEOMETRY TOPICS 13.1 Binary Space-Partitioning Trees in 2D BSP Tree Representation of a Polygon Minimum Splits versus Balanced Trees Point in Polygon Using BSP Trees Partitioning a Line Segment by a BSP Tree 13.2 Binary Space-Partitioning Trees in 3D BSP Tree Representation of a Polyhedron Minimum Splits versus Balanced Trees Point in Polyhedron Using BSP Trees Partitioning a Line Segment by a BSP Tree Partitioning a Convex Polygon by a BSP Tree 13.3 Point in Polygon Point in Triangle Point in Convex Polygon Point in General Polygon Faster Point in General Polygon A Grid Method 13.4 Point in Polyhedron Point in Tetrahedron Point in Convex Polyhedron Point in General Polyhedron 13.5 Boolean Operations on Polygons The Abstract Operations The Two Primitive Operations Boolean Operations Using BSP Trees Other Algorithms 13.6 Boolean Operations on Polyhedra Abstract Operations Boolean Operations Using BSP Trees 13.7 Convex Hulls Convex Hulls in 2D

12 Contents XIX Convex Hulls in 3D Convex Hulls in Higher Dimensions 13.8 Delaunay Triangulation Incremental Construction in 2D Incremental Construction in General Dimensions Construction by Convex Hull 13.9 Polygon Partitioning Visibility Graph of a Simple Polygon Triangulation Triangulation by Horizontal Decomposition Convex Partitioning Circumscribed and Inscribed Balls Circumscribed Ball Inscribed Ball Minimum Bounds for Point Sets Minimum-Area Rectangle Minimum-Volume Box Minimum-Area Circle Minimum-Volume Sphere Miscellaneous Area and Volume Measurements Area ofa2d Polygon Area of a 3D Polygon Volume of a Polyhedron APPENDIX 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

13 xx Contents A.4 Representations of 3D Rotations 857 A.4.1 Matrix Representation 857 A.4.2 Axis-Angle Representation 858 A.4.3 Quaternion Representation 860 A.4.4 Performance Issues 861 A. 5 Root Finding 869 A.5.1 Methods in One Dimension 869 A.5.2 Methods in Many Dimensions 874 A.5.3 Stable Solution to Quadratic Equations 875 A.6 Minimization 876 A.6.1 Methods in One Dimension 876 A.6.2 Methods in Many Dimensions 877 A.6.3 Minimizing a Quadratic Form 880 A.6.4 Minimizing a Restricted Quadratic Form 880 A. 7 Least Squares Fitting 882 A.7.1 Linear Fitting of Points (x,f(x)) 882 A.7.2 Linear Fitting of Points Using Orthogonal Regression 882 A.7.3 Planar Fitting of Points (x,y,f(x, y)) 884 A.7.4 Hyperplanar Fitting of Points Using Orthogonal Regression 884 A.7.5 Fitting a Circle to 2D Points 886 A.7.6 Fitting a Sphere to 3D Points 887 A.7.7 Fitting a Quadratic Curve to 2D Points 888 A.7.8 Fitting a Quadric Surface to 3D Points 889 A.8 Subdivision of Curves 889 A.8.1 Subdivision by Uniform Sampling 889 A.8.2 Subdivision by Arc Length 890 A.8.3 Subdivision by Midpoint Distance 891 A.8.4 Subdivision by Variation 892 A. 9 Topics from Calculus 894 A.9.1 Level Sets 894 A.9.2 Minima and Maxima of Functions 898 A.9.3 Lagrange Multipliers 910 APPENDIX B TRIGONOMETRY B. 1 Introduction B. 1.1 Terminology B.1.2 Angles B.I.3 Conversion Examples