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## Contents

• 1. Representation of Curves and Surfaces
• 1.1 Analytic representation of curves
• 1.1.1 Plane curves
• 1.1.2 Space curves
• 1.2 Analytic representation of surfaces
• 1.3 Bézier curves and surfaces
• 1.3.1 Bernstein polynomials
• 1.3.2 Arithmetic operations of polynomials in Bernstein form
• 1.3.3 Numerical condition of polynomials in Bernstein form
• 1.3.4 Definition of Bézier curve and its properties
• 1.3.5 Algorithms for Bézier curves
• 1.3.6 Bézier surfaces
• 1.4 B-spline curves and surfaces
• 1.4.1 B-splines
• 1.4.2 B-spline curve
• 1.4.3 Algorithms for B-spline curves
• 1.4.4 B-spline surface
• 1.5 Generalization of B-spline to NURBS

• 2. Differential Geometry of Curves
• 2.1 Arc length and tangent vector
• 2.2 Principal normal and curvature
• 2.3 Binormal vector and torsion
• 2.4 Frenet-Serret formulae

• 3. Differential Geometry of Surfaces
• 3.1 Tangent plane and surface normal
• 3.2 First fundamental form I (metric)
• 3.3 Second fundamental form II (curvature)
• 3.4 Principal curvatures
• 3.5 Gaussian and mean curvatures
• 3.5.1 Explicit surfaces
• 3.5.2 Implicit surfaces
• 3.6 Euler's theorem and Dupin's indicatrix

• 4. Nonlinear Polynomial Solvers and Robustness Issues
• 4.1 Introduction
• 4.2 Local solution methods
• 4.3 Classification of global solution methods
• 4.3.1 Algebraic and Hybrid Techniques
• 4.3.2 Homotopy (Continuation) Methods
• 4.3.3 Subdivision Methods
• 4.4 Projected Polyhedron algorithm
• 4.5 Auxiliary variable method for nonlinear systems with square roots of polynomials
• 4.6 Robustness issues
• 4.7 Interval arithmetic
• 4.8 Rounded interval arithmetic and its implementation
• 4.8.1 Double precision floating point arithmetic
• 4.8.2 Extracting the exponent from the binary representation
• 4.8.3 Comparison of two different unit-in-the-last-place implementations
• 4.8.4 Hardware rounding for rounded interval arithmetic
• 4.8.5 Implementation of rounded interval arithmetic
• 4.9 Interval Projected Polyhedron algorithm
• 4.9.1 Formulation of the governing polynomial equations
• 4.9.2 Comparison of software and hardware rounding

• 5. Intersection Problems
• 5.1 Overview of intersection problems
• 5.2 Intersection problem classification
• 5.2.1 Classification by dimension
• 5.2.2 Classification by type of geometry
• 5.2.3 Classification by number system
• 5.3 Point/point intersection
• 5.4 Point/curve intersection
• 5.4.1 Point/implicit algebraic curve intersection
• 5.4.2 Point/rational polynomial parametric curve intersection
• 5.4.3 Point/procedural parametric curve intersection
• 5.5 Point/surface intersection
• 5.5.1 Point/implicit algebraic surface intersection
• 5.5.2 Point/rational polynomial parametric surface intersection
• 5.5.3 Point/procedural parametric surface intersection
• 5.6 Curve/curve intersection
• 5.6.1 Rational polynomial parametric/implicit algebraic curve intersection (Case D3)
• 5.6.2 Rational polynomial parametric/rational polynomial parametric curve intersection (Case D1)
• 5.6.3 Rational polynomial parametric/procedural parametric and procedural parametric/procedural parametric curve intersections (Cases D2 and D5)
• 5.6.4 Procedural parametric/implicit algebraic curve intersection (Case D6)
• 5.6.5 Implicit algebraic/implicit algebraic curve intersection (Case D8)
• 5.7 Curve/surface intersection
• 5.7.1 Rational polynomial parametric curve/implicit algebraic surface intersection (Case E3)
• 5.7.2 Rational polynomial parametric curve/rational polynomial parametric surface intersection (Case E1)
• 5.7.3 Rational polynomial parametric/procedural parametric and procedural parametric/procedural parametric curve/surface intersections (Cases E2/E6)
• 5.7.4 Procedural parametric curve/implicit algebraic surface intersection (Case E7)
• 5.7.5 Implicit algebraic curve/implicit algebraic surface intersection (Case E11)
• 5.7.6 Implicit algebraic curve/rational polynomial parametric surface intersection (Case E9)
• 5.8 Surface/surface intersections
• 5.8.1 Rational polynomial parametric/implicit algebraic surface intersection (Case F3)
• 5.8.2 Rational polynomial parametric/rational polynomial parametric surface intersection (Case F1)
• 5.8.3 Implicit algebraic/implicit algebraic surface intersection (Case F8)
• 5.9 Overlapping of curves and surfaces
• 5.10 Self-intersection of curves and surfaces
• 5.11 Summary

• 6. Differential Geometry of Intersection Curves
• 6.1 Introduction
• 6.2 More differential geometry of curves
• 6.3 Transversal intersection curve
• 6.3.1 Tangential direction
• 6.3.2 Curvature and curvature vector
• 6.3.3 Torsion and third order derivative vector
• 6.3.4 Higher order derivative vector
• 6.4 Intersection curve at tangential intersection points
• 6.4.1 Tangential direction
• 6.4.2 Curvature and curvature vector
• 6.4.3 Third and higher order derivative vector
• 6.5 Examples
• 6.5.1 Transversal intersection of parametric-implicit surfaces
• 6.5.2 Tangential intersection of implicit-implicit surfaces

• 7. Distance Functions
• 7.1 Introduction
• 7.2 Problem formulation
• 7.2.1 Definition of the distances between two point sets
• 7.2.2 Geometric interpretation of stationarity of distance function
• 7.3 More about stationary points
• 7.3.1 Classification of stationary points
• 7.3.2 Nonisolated stationary points
• 7.4 Examples

• 8. Curve and Surface Interrogation
• 8.1 Classification of interrogation methods
• 8.1.1 Zeroth-order interrogation methods
• 8.1.2 First-order interrogation methods
• 8.1.3 Second-order interrogation methods
• 8.1.4 Third-order interrogation methods
• 8.1.5 Fourth-order interrogation methods
• 8.2 Stationary points of curvature of free-form parametric surfaces
• 8.2.1 Gaussian curvature
• 8.2.2 Mean curvature
• 8.2.3 Principal curvatures
• 8.3 Stationary points of curvature of explicit surfaces
• 8.4 Stationary points of curvature of implicit surfaces
• 8.5 Contouring constant curvature
• 8.5.1 Contouring levels
• 8.5.2 Finding starting points
• 8.5.3 Mathematical formulation of contouring
• 8.5.4 Examples

• 9. Umbilics and Lines of Curvature
• 9.1 Introduction
• 9.2 Lines of curvature near umbilics
• 9.3 Conversion to Monge form
• 9.4 Integration of lines of curvature
• 9.5 Local extrema of principal curvatures at umbilics
• 9.6 Perturbation of generic umbilics
• 9.7 Inflection lines of developable surfaces
• 9.7.1 Differential geometry of developable surfaces
• 9.7.2 Lines of curvature near inflection lines

• 10. Geodesics
• 10.1 Introduction
• 10.2 Geodesic equation
• 10.2.1 Parametric surfaces
• 10.2.2 Implicit surfaces
• 10.3 Two point boundary value problem
• 10.3.1 Introduction
• 10.3.2 Shooting method
• 10.3.3 Relaxation method
• 10.4 Initial approximation
• 10.4.1 Linear approximation
• 10.4.2 Circular arc approximation
• 10.5 Shortest path between a point and a curve
• 10.6 Numerical applications
• 10.6.1 Geodesic path between two points
• 10.6.2 Geodesic path between a point and a curve
• 10.7 Geodesic offsets
• 10.8 Geodesics on developable surfaces

• 11. Offset Curves and Surfaces
• 11.1 Introduction
• 11.1.1 Background and motivation
• 11.1.2 NC machining
• 11.1.3 Medial axis
• 11.1.4 Tolerance region
• 11.2 Planar offset curves
• 11.2.1 Differential geometry
• 11.2.2 Classification of singularities
• 11.2.3 Computation of singularities
• 11.2.4 Approximations
• 11.3 Offset surfaces
• 11.3.1 Differential geometry
• 11.3.2 Singularities of offset surfaces
• 11.3.3 Self-intersection of offsets of implicit quadratic surfaces
• 11.3.4 Self-intersection of offsets of explicit quadratic surfaces
• 11.3.5 Self-intersection of offsets of polynomial parametric surface patches
• 11.3.6 Tracing of self-intersection curves
• 11.3.7 Approximations
• 11.4 Pythagorean hodograph
• 11.4.1 Curves
• 11.4.2 Surfaces
• 11.5 General offsets
• 11.6 Pipe surfaces
• 11.6.1 Introduction
• 11.6.2 Local self-intersection of pipe surfaces
• 11.6.3 Global self-intersection of pipe surfaces

• Problems
• A. Color Plates
• Bibliography
• Index
• Software Package

December 2009