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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
- 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
December 2009
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