Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable

Transcription

1 Rida T. Farouki Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable With 204 Figures and 15 Tables 4y Springer

2 Contents 1 Introduction The Lure of Analytic Geometry Symbiosis of Algebra and Geometry Computer-aided Geometric Design Pythagorean-hodograph Curves Algorithms and Applications 7 Part I Algebra 2 Preamble A Historical Enigma Theorem of Pythagoras Al-Jabr wa'1-muqabala Fields, Rings, and Groups 25 3 Polynomials Basic Properties Polynomial Bases Roots of Polynomials Resultants and Discriminants Rational Functions 41 4 Complex Numbers Caspar Wessel Elementary Properties Functions of Complex Variables Differentiation and Integration Geometry of Conformal Maps Harmonic Functions 56

3 VIII Contents 4.7 Conformal Transplants Some Simple Mappings 58 5 Quaternions Multi-dimensional Numbers No Three-dimensional Numbers Sums and Products of Quaternions Quaternions and Spatial Rotations Rotations as Products of Reflections Families of Spatial Rotations Four-dimensional Rotations 74 6 Clifford Algebra Clifford Algebra Bases Algebra of Multivectors The Geometric Product Reflections and Rotations 85 Part II Geometry 7 Coordinate Systems Cartesian Coordinates Barycentric Coordinates Barycentric Coordinates on Intervals Barycentric Coordinates on Triangles Transformation of the Domain Barycentric Points and Vectors Directional Derivatives Polynomial Bases Over Triangles Un-normalized Barycentric Coordinates Three or More Dimensions Curvilinear Coordinates One-to-one Correspondence Distance and Angle Measurements Jacobian of the Transformation Example: Plane Polar Coordinates Three or More Dimensions Ill 7.4 Homogeneous Coordinates The Projective Plane Circular Points and Isotropic Lines The Principle of Duality Projective Transformations Invariance of the Cross Ratio Geometrical Figures and their Shadows Projective Geometry of Three Dimensions 127

4 Contents IX Differential Geometry Intrinsic Geometry of Plane Curves Tangent and Curvature The Circle of Curvature Vertices of Plane Curves The Intrinsic Equation Families of Plane Curves Envelopes of Curve Families Families of Implicit Curves Families of Parametric Curves Families of Lines and Circles Evolutes, Involutes, Parallel Curves Tangent Line and Osculating Circle Evolutes and Involutes The Horologium Oscillatorium Families of Parallel (Offset) Curves Trimming the Untrimmed Offset Intrinsic Geometry of Space Curves Curvature and Torsion The Frenet Frame Inflections of Space Curves Intrinsic Equations Intrinsic Geometry of Surfaces First Fundamental Form Second Fundamental Form Curves Lying on a Surface Normal Curvature of a Surface Principal Curvatures and Directions Local Surface Shape Gauss Map of a Surface Lines of Curvature Geodesies on a Surface 193 Algebraic Geometry Parametric and Implicit Forms Plane Algebraic Curves Singular Points Intersections with a Straight Line Double Points of Algebraic Curves Higher-order Singular Points Genus of an Algebraic Curve Resolution of Singularities Birational Transformations.' Pliicker Relations 212

5 X Contents Bezout's Theorem Implicitization and Parameterization Algebraic Surfaces Singular Points and Curves Rationality of Algebraic Surfaces Algebraic Space Curves Composite Surface Intersections Plane Projections of a Space Curve Genus of an Algebraic Space Curve Singularities of Space Curves Non-Euclidean Geometry The Metric Tensor Contravariant and Covariant Vectors Methods of Tensor Algebra The Geodesic Equations Differentiation of Tensors Parallel Transport of Vectors 243 Part III Computer Aided Geometric Design 11 The Bernstein Basis Theorem of Weierstrass Bernstein-form Properties The Control Polygon Transformation of Domain Degree Operations de Casteljau Algorithm Arithmetic Operations Computing Roots on (0,1) Numerical Condition Numerical Stability License to Compute Characterization of Errors Floating-point Computations Floating-point Numbers Floating-point Arithmetic Dangers of Digit Cancellation Models for Error Propagation Stability and Condition Numbers Condition of a Polynomial Value Condition of a Polynomial Root Wilkinson's Polynomial 277

6 Contents Vector and Matrix Norms Condition of a Linear Map Basis Transformations Subdivision Processes Ill-posed Problems Backward Error Analysis Equivalent Input Errors Example: Horner's Method Bezier Curves and Surfaces Convex-hull Confinement Variation-diminishing Property Degree Elevation de Casteljau Algorithm Bezier Curve Hodographs Rational Bezier Curves Conies as Bezier Curves Tensor-product Surface Patches Triangular Surface Patches C 2 Cubic Spline Curves Mechanical Splines Elastic Bending Energy Polynomial Interpolation The Lagrange Basis Convergence Behavior C 2 Cubic Spline Functions Cubic Hermite Form C 2 Continuity Equations Choice of End Conditions Solution of Tridiagonal Systems Minimum Energy Property Spline Approximation Convergence C 2 Cubic Spline Curves Choice of Knot Sequence Parametric or Geometric Continuity Geometric Hermite Interpolation Elastica or "Non-linear" Splines Spline Basis Functions Bases for Spline Functions ; The Cardinal Basis Construction of Cardinal Basis Bivariate Spline Functions Tensor-product Spline Surfaces 351 XI

7 XII Contents 15.3 The B-spline Basis The Knot Vector Cox-de Boor Algorithm Tensor-product B-spline Surfaces Rational B-spline Curves and Surfaces Bezier and B-spline Forms Compared Spline Basis Conversion Cardinal to B-spline Form Basis Conversion Matrix 365 Part IV Planar Pythagorean hodograph Curves 16 Arc-length Parameterization In Search of an Elusive Ideal The Rectification of Curves Polynomial Parametric Speed Algebraically-rectifiable Curves :5 Unit Speed Approximations Pythagorean hodograph Curves Planar Pythagorean Hodographs Bezier Control Points of PH Curves Parametric Speed and Arc Length Differential and Integral Properties Rational Offsets of PH Curves Tschirnhausen's Cubic Ehrenfried Walther von Tschirnhaus Tschirnhaus and Caustic Curves Unique Pythagorean-hodograph Cubic You Mean we Pay you to do ThatW Complex Representation Complex Curves and Hodographs One-to-one Correspondence Rotation Invariance of Hodographs Pythagorean-hodograph Cubics Revisited Characterization of the PH Quintics Geometry of the Control Polygon Intrinsic Features of Corresponding Curves 422

8 Contents XIII 20 Rational Pythagorean-hodograph Curves Construction of Rational PH Curves Dual Bezier Curve Representation Relation to Polynomial PH Curves Rational Arc Length Functions Geometrical Optics Interpretation Laguerre Geometry Formulation Improper Rational Parameterizations Rational Surfaces with Rational Offsets Minkowski Isoperimetric-hodograph Curves 451 Part V Spatial Pythagorean hodograph Curves 21 Pythagorean Hodographs in R Geometry of Spatial PH Cubics Spatial Pythagorean Hodographs Bezier Control Polygons Differential Properties Quaternion Representation Pythagorean Condition in K Degeneration of Spatial PH Curves Rotation Invariance of Hodographs Reflection Form of Hodographs One-to-one Correspondence? Helical Polynomial Curves Helical Curves and PH Curves? Morphology of Helical PH Quintics Monotone-helical PH Quintics General Helical PH Quintics Sufficient and Necessary Conditions Minkowski Pythagorean Hodographs The Minkowski Metric Medial Axis Transform Minkowski PH Curves in K Clifford Algebra Representation MAT Approximation by MPH Curves Generalization to the Space M 3 ' 1 519

9 XIV Contents Part VI Algorithms 25 Planar Hermite Interpolants Hermite Interpolation Problem Solution in Complex Representation The Absolute Rotation Index Comparison with "Ordinary" Cubics Higher-order Hermite Interpolants Monotone Curvature Segments Elastic Bending Energy Complex Form of the Integrand Energy of Tschirnhaus Segments Bending Energy of PH Quintics The "Gracefulness" of PH Quintics Minimal-energy Hermite Interpolants Planar C 2 PH Quintic Splines Construction of PH Splines C 2 PH Quintic Spline Equations End Conditions for PH Splines Number of Distinct Interpolants Solution by Homotopy Method Choice of Initial System Predictor-corrector Procedure Empirical Results and Examples Solution by Iterative Methods Choice of Starting Approximation Functional Iteration and Relaxation Newton-Raphson Method Computed Examples Generalizations of PH Splines Non-uniform Knot Sequences Shape-preserving PH Splines Control Polygons for PH Splines Equivalent Interpolation Problem Inclusion of Multiple Knots Emulating B-spline Curve Properties Illustrative Examples Spatial Hermite Interpolants G l Interpolation by Cubics C 1 Hermite Interpolation Problem Rotation Invariance of Interpolants Residual Degrees of Freedom. : 600

10 Contents 28.5 Integral Measures of Shape Clifford Algebra Formulation Helical PH Quintic Interpolants Higher-order Hermite Interpolants Spatial C 2 PH Quintic Splines 613 XV Part VII Applications 29 Real-time CNC Interpolators Digital Motion Control Taylor Series Interpolators PH Curve Interpolators Constant Feedrate Curvature-dependent Feedrate Offset Curve Interpolator Feedrate in Terms of Arc Length Linear Dependence on Arc Length Quadratic Dependence on Arc Length Time-dependent Feedrate Polynomial Time Dependence Acceleration/Deceleration Profiles Traversing a Single PH Curve Experimental Results Constant Material Removal Rate Form of Feedrate Function Interpolator Algorithm Experimental Results Contour Machining of Surfaces Tool Path Generation Optimal Contour Orientations Rotation minimizing Frames Introduction and Motivation Adapted Frames on Space Curves Euler-Rodrigues Frame for PH Curves Rotation-minimizing Frames Energy of Framed Space Curves Exact RMFs on PH Curves Integration of Rational Functions Frames for PH Cubics and Quintics Rational RMF Approximations Rational Hermite Interpolation Computed Examples Parameterization of Canal Surfaces 689

11 XVI Contents 31 Closure 693 References 697 Index 719