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Shape Interrogation for Computer Aided Design and Manufacturing
[size=-1](Hyperbook Edition)
Please mail to wjcho@mit.edu for errata.

[size=+1]Nicholas M. Patrikalakis      [size=+1]Takashi Maekawa      [size=+1]Wonjoon Cho

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• Preface
• 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
      

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December 2009

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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
5.4.2.1 Elementary method
5.4.2.2 Bounding box and subdivision followed by minimization method
5.4.2.3 Distance function method
5.4.2.4 Implicitization
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
5.5.2.1 Implicitization
5.5.2.2 Newton's method
5.5.2.3 Bounding box and subdivision followed by minimization method
5.5.2.4 Distance function method
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)
5.6.1.1 2-D planar case
5.6.1.2 3-D space curve
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)
5.6.5.1 Implicitization
5.6.5.2 Newton's method
5.6.5.3 Interval Projected Polyhedron solver
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)
5.7.2.1 Implicitization
5.7.2.2 Bounding box and subdivision followed by minimization method
5.7.2.3 Interval Projected Polyhedron solver
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)
5.8.1.1 Formulation
5.8.1.2 Tracing method
5.8.1.3 Characteristic points
5.8.1.4 Analysis of singular points
5.8.1.5 Computing starting points for all branches
5.8.2 Rational polynomial parametric/rational polynomial parametric surface intersection (Case F1)
5.8.2.1 Lattice methods
5.8.2.2 Subdivision methods
5.8.2.3 Marching methods
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
6.4.3.1 Parametric-parametric
6.4.3.2 Implicit-implicit
6.4.3.3 Parametric-implicit
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
8.1.1.1 Wireframe
8.1.1.2 Contouring
8.1.2 First-order interrogation methods
8.1.2.1 Shading and ray tracing
8.1.2.2 Isophotes
8.1.2.3 Reflection lines
8.1.2.4 Highlight lines
8.1.3 Second-order interrogation methods
8.1.3.1 Curvature plots
8.1.3.2 Zero curvature points
8.1.3.3 Radial curves
8.1.3.4 Surface curvatures and curvature maps
8.1.3.5 Focal curves and surfaces
8.1.3.6 Orthotomics
8.1.3.7 Curvature lines
8.1.3.8 Geodesics
8.1.4 Third-order interrogation methods
8.1.4.1 Torsion of space curves
8.1.4.2 Stationary points of curvature of planar and space curves
8.1.4.3 Stationary points of curvature of parametric surfaces
8.1.5 Fourth-order interrogation methods
8.1.5.1 Stationary points of torsion of space curve
8.1.5.2 Stationary points of total curvature
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
8.5.4.1 Gaussian curvature
8.5.4.2 Mean curvature
8.5.4.3 Maximum principal curvature
8.5.4.4 Minimum principal curvature

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
11.1.3.1 Algorithms for determining the MAT or related sets
11.1.3.2 Theoretical analysis of the MAT properties
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
11.6.3.1 End circle to end circle global self-intersection
11.6.3.2 Body to body global self-intersection
11.6.3.3 End circle to body global self-intersection
11.6.3.4 A necessary and sufficient condition for nonsingularity

Problems
A. Color Plates
Bibliography
Index
Software Package

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