| 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 |
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