Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry Representation Ab initio design Rendering Solid modelers Kinematic analysis History p. 6 USAF SAGE Sketchpad Sutherland Ross APT Coons Ferguson Casteljua Bezier Eshleman Meriwether Forrest Sculptured surfaces Gordon Riesenfeld Barsky Requicha Voelker The Mathematics p. 11 Linear algebra Vectors Matrix methods Set theory Boolean algebra Polynomial interpolation Numerical methods Newton's method Quadrature Vectors p. 15
Basis vectors Free vector Position vector Tensor form Vector magnitude Vector direction Unit vector Vector addition Scalar product Vector product Solution of vector equations Summary of vector properties Matrices p. 23 Row matrix Column matrix Square matrix Identity matrix Scalar matrix Diagonal matrix Symmetric matrix Kronecker delta Matrix transpose Matrix addition Scalar multiplication Matrix multiplication Partitioned matrices Matrix inversion Scalar and vector products Eigenvalues and eigenvectors Summary of matrix properties Determinants Determinant properties Conventions and Notation p. 30 For scalars Vectors Matrices Determinants Quaternions Boolean operators Intervals Differential equations
Hermite curve Bezier curve B-spline curve Hermite surface Bezier surface B-spline surface Curves p. 34 Intrinsic Equations of Curves p. 34 Torsion Arc length Natural equations Curvature Curvature functions Explicit and Implicit Equations of Curves p. 37 Closed and open curves Axis dependent Symmetries of plane curves Parametric Equations of Curves p. 38 Parametric equations Parametric variable Curve segment Bounding points Closed interval Domain and normalization Model space Parameter space Reparameterization Separation of variables Conic Curves p. 48 Second degree implicit equation Standard form Conic curve characteristics Parametric forms: parabola, hyperbola, ellipse Points on a Curve p. 48 Direct point solution Inverse point solution Horner's Rule Forward difference method Vector interpretation of the inverse point solution Hermite, Bezier, and B-Spline Curves: An Overview p. 53 Hermite Curves p. 56
Algebraic and Geometric Forms p. 56 Algebraic coefficients Power basis representation Differentiation Geometric coefficients Hermite basis functions Hermite Basis Functions p. 60 Universality Dimensional independence Separation of boundary coefficient effects Curve decomposition Orthogonal curves Domain of the parametric variable First derivative basis function curves Matrix Form p. 66 Matrix of algebraic coefficients Hermite geometry matrix Hermite basis transformation matrix Tangent Vectors p. 69 Parameterization effect on tangent vector magnitude Direction co-sines 12 Degrees of freedom Effect on curve shape Truncating and Subdividing p. 75 Direction of parameterization Reparameterization Truncating a parametric curve Preserving the degree of the parametric polynomial Curve segments Transformed algebraic and geometric coefficients Renormalization Three-Point Interpolation p. 81 Derivation and constraints Four-Point Interpolation p. 84 The four point form Equal partition of unity characteristic Conic Hermite Curves p. 87 The classical construction a conic curve The Hermite approximation of the construction Representing a parabola Representing a hyperbola
Representing a circular arc Deviation Composite Hermite Curve p. 99 Blending a curve between two disjoint curves Parametric continuity Geometric continuity Lagrange and Hermite interpolation The spline curve The second-derivative form of a cubic Hermite curve Equations describing the elastic deformation of a beam Derivation of the composite Hermite spline curve Transformation from a local to a global coordinate system Bezier Curves p. 122 Bezier Basis Functions p. 123 Bernstein polynomials Binomial coefficient Matrix form Bezier basis transformation matrix Bezier-Hermite conversion Invariance under affine transformations Bezier curve by geometric construction Control Points p. 130 Control polygon Convex hull Partition-of-unit property Modifying Bezier curves Closed curves Continuity Direction of parameterization Degree elevation Truncating and Subdividing p. 135 Truncating a second-degree curve Reparameterization Truncating a cubic curve Recursive subdivision Subdivision by geometric construction Composite Bezier Curves p. 142 Determining continuity conditions Rational Bezier Curves p. 144 Homogeneous coordinates Four-dimensional projective space
Homogeneous coordinate geometry in one and two dimensions Control points and weights Transformation invariance considerations B-Spline Curves p. 149 Nonuniform B-Spline Basis Functions p. 149 Nonrational form Knot values Basis functions Recursive definition Coincident knot values Basis function independence Matrix form Uniform B-Spline Basis Functions p. 160 Periodic basis functions Uniform knot vector Interpolating control points Quadratic and Cubic B-Spline Basis Functions p. 163 Comparing to Hermite and Bezier basis functions Closed B-Spline Curves p. 164 Modifying segment number range Control points Continuity p. 167 Continuity between segments of a B-spline curve Parametric continuity Control point multiplicity Conversion between Basis Functions p. 169 From B-spline to Bezier From B-spline to Hermite Vice versa Nonuniform Rational B-Spline Curves p. 171 The vector-valued piecewise rational polynomial Representing Conics with NURBS Curves p. 173 The second-degree NURBS curve The seven-point square-based NURBS circle Cubic Beta Splines p. 174 A variation on the B-spline Adding global control over curve shape Bias and tension Surfaces p. 177 Explicit and Implicit Equations of Surfaces p. 177 Implicit equations
The unbounded plane Quadric surface Sphere Cylinder Explicit equations Implicitization The tensor product Quadric Surfaces p. 179 Quadric equation coefficients Signature Quadric surface of revolution Canonical equation Parametric Equation of Surfaces p. 182 The surface patch Tangent and twist vectors A plane Sphere Ellipsoid Surface of revolution Parameter space of a surface Points on a Surface p. 189 Point evaluation Inverse point solution Changing the parametric net Curve Nets p. 191 Parametric Orthogonal Conjugate nets Isoparametric curves Covariant net Embedded Curves p. 192 Curves on surfaces Irregular patch boundary curves Decomposing a complex shape Trimmed patch Point classification Halfspaces Hermite, Bezier, and B-Spline Surfaces: An Overview p. 197 Bicubic Hermite Surfaces p. 200 Algebraic and Geometric Forms p. 200 Algebraic coefficients
Tensor product Parameterization Matrix notation Tangent and twist vectors Geometric interpretation of twist vectors Mutually orthogonal nets of parametric curves Boundary curves Boundary conditions Auxiliary curves Evaluating a point on a patch Hermite Patch Basis Functions p. 212 Basis functions Tangent vector basis functions Tangent and Twist Vectors p. 212 Mixed partial derivatives Continuity considerations The F patch of zero twist vectors Effect of twist vectors on the patch interior Normals p. 214 Unit normal Normal vector Sign convention for normal direction 16-Point Form p. 217 The matrix form Point distribution over a patch The four-curve form: a variation on the 16-point form Reparameterization of a Patch p. 221 Reverse parameterization Affect on patch normals General reparameterization Parameterization of a rectangular array of patches Truncating and Subdividing a Patch p. 225 Reparameterizing Computing tangent and twist vectors Composite Hermite Surfaces p. 227 Continuity Shape control Effect of twist vectors on continuity Degrees of freedom available for shaping Basis function invariance Relationship between adjacent auxiliary curves
Indexing schemes for patch arrays Distribution of scale factors across patch boundaries Parametric spline interpolation Rectangular network of intersecting curves Normalizing the parametric variables Cardinal spline-interpolating function Mesh points Transition from a complex to a simple cross section Special Hermite Patches p. 238 Plane surface Special form of a Hermite plane patch Cylindrical surface Ruled surface Degenerate patches Blend Surfaces p. 251 Blend between two disjoint patches Blend to the boundaries of another patch General blend surfaces Bezier Surfaces p. 255 The Tensor Product Bezier Patch p. 255 Rectangular array of control points Tensor product form Characteristic polyhedron Convex hull Basis functions Matrix form The Bicubic Bezier Patch p. 256 Control points Matrix form Points defining the characteristic polyhedron Boundary curves Control points influencing the twist vector at a patch corner A 3x5 Rectangular Array of Control Points p. 259 Advantage of a five-point boundary curve Converting between Bicubic Bezier and Hermite Forms p. 260 Mathematical equivalence of the forms Degree Elevation in a Bezier Surface p. 261 Manipulating the shape of a surface Adding control points Composite Bezier Surfaces p. 262 Geometric continuity
Common boundary curves Rational Bezier Patch p. 263 Properties and effects of weights B-Spline Surfaces p. 265 The Tensor Product B-Spline Surface p. 265 Matrix Form p. 265 Open and Closed B-Spline Surfaces p. 266 Open quadric surface idealization Open cubic surface idealization Open quartic surface idealization Open quintic surface idealization Open cubic-quadric surface idealizations Partially closed surface Nonuniform Rational B-Spline Surfaces p. 271 Solids p. 275 Parametric Solids p. 275 The hyperpatch Trivariate parametric functions Isoparametric surface Boundary elements Corner points Edge curves Bounding faces The tricubic Hermite solid The Tricubic Solid p. 278 Algebraic coefficients Hermite basis functions Boundary condition array Indexing schemes Boundary condition vectors Geometric coefficients Triple mixed partial derivative terms Contracting indices Tangent vectors Twist vectors Continuity and composite solids Curves and Surfaces Embedded in a Solid p. 291 Isoparametric curves and surfaces Continuity conditions Parametric cell Orthogonal cells
Orthogonal parametric curve nets Nonisoparametric curve Curvilinear coordinate system Trimmed boundaries Generalized Notation Scheme and Higher-Dimension Elements p. 295 A generalized notation and summation scheme Indices and subscript interpretation Instances and Parameterized Shapes p. 298 Primitive shape Uniform and differential scaling Group technology Sweep Solids p. 302 Translational sweep Extrusion General sweep Generator Director The position-direction (PD) curve Constant and variable cross section solids Rotational sweeps Profile curve Parallels Surface of revolution Controlled Deformation Solids p. 313 Nonlinear transformations Curvilinear coordinate system Basis deformation Axial deformations Bivariate deformation Trivariate deformation Deformable surfaces Complex Model Construction p. 318 Topology of Models p. 318 Piecewise flat surfaces Euler's formula Determining all possible regular polyhedra Polytopes Nonsimple polyhedra Connectivity number Genus The Euler-Poincare formula
Topological atlas Orientation Nonorientable surfaces Mobius strip Klein bottle Handles Topological equivalence Transition parity Curvature of piecewise flat surfaces The Euler characteristic of a surface Topology of closed curved surfaces Gauss-Bonnet theorem Euler operators Euler object Topological disks Nets Graph-Based Models p. 335 Nodes and branches Connectivity matrix Adjacency matrix Directed graph In degree Out degree Circuit Tree Subgraph Spanning tree Leaf node Root Depth Binary tree Traversals Boolean Models p. 342 Set theory Set-builder notation Elements Venn diagrams Union Intersection Difference Properties of operations on sets
Open and closed sets Set membership classification Winding number Inside-outside classification Dimensional homogeneity Regularized set operators Boolean operators Order dependence Boundary test Boolean Model Construction p. 364 Boolean model Procedural models Unevaluated model Halfspace Union Intersection Difference Constructive Solid Geometry p. 370 Binary tree model representation Primitive solids Primitives as intersections of halfspaces Boundary evaluation T-edges Neighborhood model Combining neighborhood models Boundary Models p. 377 B-rep model Generalized concept of a boundary Face boundary convention Geometric Properties p. 387 Local Properties of a Curve p. 387 Tangent vector and line Normal plane Principal normal vector and line Binormal vector Osculating plane Rectifying plane Moving trihedron Curvature and torsion Inflection points Global Properties of a Curve p. 401
Arc length Gaussian quadrature Characteristic tests Loops Cusps Local Properties of a Surface p. 404 Fundamental forms Normal to a surface Tangent plane Principal curvature Normal curvature Principal normal curvatures Umbilical point Geodesics Geodesic curvature Properties of curves on surfaces Point classification Osculating paraboloid Elliptic point Hyperbolic point Parabolic point Planar point Global Properties of a Surface p. 415 Surface area Gaussian curvature (again) Volume Characteristic tests (planar, spherical, developable) Global Properties of Complex Solids p. 417 Representation-dependent methods The Timmer-Stern method Spatial enumeration by point classification A cell-partitioned solid Block decomposition by cell classification Relational Properties p. 429 Minimum distance between two points Minimum distance between a point and a curve Minimum distance between a point and a surface Minimum distance between two curves Minimum distance between two surfaces Nearest neighbor spatial search Intersections p. 442
Intersections with straight lines Plane intersections Curve intersections Surface intersections The hunting phase The tracing phase The ordering phase Computation of parametric derivatives Surface inversion Surface-surface intersection Step-size selection for the tracing phase Answers to Selected Exercises p. 467 Index p. 497 Table of Contents provided by Blackwell's Book Services and R.R. Bowker. Used with permission.