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Curves and Surfaces.

By: Laurent, Pierre-Jean.
Contributor(s): Méhauté, Alain Le | Schumaker, Larry L.
Material type: TextTextSeries: eBooks on Demand.Publisher: Kent : Elsevier Science & Technology, 2014Copyright date: ©1991Description: 1 online resource (535 pages).Content type: text Media type: computer Carrier type: online resourceISBN: 9781483263878.Subject(s): Approximation theory -- Congresses | Curves -- Computer simulation -- Congresses | Spline theory -- Congresses | Surfaces -- Computer simulation -- CongressesGenre/Form: Electronic books.Additional physical formats: Print version:: Curves and SurfacesDDC classification: 511.4 Online resources: Click here to view this ebook.
Contents:
Front Cover -- Curves and Surfaces -- Copyright Page -- Table of Contents -- PREFACE -- CONTRIBUTORS -- Chapter 1. Parametrization for Data Approximation -- 1. Introduction -- 2. Local Taylor Expansion of the Curve -- 3. Criteria -- 4. Application to Approximation with B-Splines -- References -- Chapter 2. A Vector Spline Approximation With Application to Meteorology -- 1. Introduction -- 2. The Minimization Problem -- 3. Solution of P α,β -- 4. Limit Problems -- 5. Numerical Results -- References -- Chapter 3. Kernel Estimation in Change-Point Hazard Rate Models -- 1. Introduction -- 2. The Model -- 3. Main Results -- 4. Simulation Results and Concluding Remarks -- References -- Chapter 4. Spline Manifolds -- 1. Introduction -- 2. Basic Tools -- 3. A Fundamental Result -- 4. Spline Manifolds -- 5. P(D)-positive Vectorial Distributions -- 6. Extension of P(D)-spline -- References -- Chapter 5. Use of Simulated Annealing to Construct Triangular Facet Surfaces -- 1. Introduction -- 2. Optimal Triangulations -- 3. Locally Optimal Triangulations and Edge Swapping -- 4. Simulated Annealing -- 5. An Example -- 6. Conclusions -- References -- Chapter 6. G1 and G2 Continuity Between (SBR) Surfaces -- 1. Geometric Framework -- 2. Analytical Conditions -- 3. Geometric Continuity Between Two Rectangular (SBR) -- References -- Chapter 7. Ray Tracing Rational Parametric Surfaces -- 1. Introduction -- 2. Implicitization -- 3. Intersection Problem -- 4. Discussion -- References -- Chapter 8. Energy-Based Segmentation of Sparse Range Data -- Abstract -- 1. Segmentation: Introduction and Background -- 2. Definition of the Model of World Surfaces -- 3. Experimentation -- References -- Chapter 9. Error Estimates for Multiquadric Interpolation -- 1. Introduction and Statement of Main Result -- 2. Proof of Theorem 1 -- References.
Chapter 10. A Geometrical Analysis for a Data Compression of 3D Anatomical Structures -- 1. Introduction -- 2. Data Format -- 3. Feature Extraction -- 4. Data Compression -- 5. Contour Reconstruction -- 6. Conclusion -- References -- Chapter 11. Ck Continuity of (SBR) Surfaces -- 1. Framework -- 2. Rectangular (SBR) Surfaces -- 3. Triangular (SBR) Surfaces -- References -- Chapter 12. A Note on Piecewise Monotonie Bivariate Interpolation -- 1. Introduction -- 2. Outline of the Algorithm -- 3. Conclusions -- References -- Chapter 13. Real-Time Signal Analysis with Quasi-Interpolatory Splines and Wavelets -- 1. Introduction -- 2. Spline Sampling of Digital Signals -- 3. Wavelet Signal Decomposition -- 4. Wavelet Signal Reconstruction -- References -- Chapter 14. Polynomial Expansions for Cardinal Interpolants and Orthonormal Wavelets -- 1. Introduction -- 2. The Case of Real Φ -- 3. The Case of Complex Φ -- 4. Construction of Quasi-Interpolation Operators -- References -- Chapter 15. Realtime Pipelined Spline Data Fitting for Sketched Curves -- 1. Introduction -- 2. Background -- 3. Data Reduction -- 4. Pipelining the Algorithm -- 6. Remarks -- References -- Chapter 16. Remarks on Digital Terrain Modelling Accuracy -- 1. On Terrain Modelling Accuracy -- 2. The Displacement-Buckling Approach -- 3. A Graphical Approach -- References -- Chapter 17. Convexity and Bernstein-Bézier Polynomials -- 1. Introduction -- 2. Some Prerequisites -- 3. Planar Curves -- 4. Convexity of Functional Bézier Surfaces -- 5. Convexity of Piecewise Polynomial Surfaces -- 6. Convexity of Parametric Bézier Patches -- 7. Converse Theorems on Convexity -- References -- Chapter 18. How to Draw a Curve Using Geometrical Data -- 1. Introduction -- 2. Algebraic Regression -- 3. Automatic Ordering -- References.
Chapter 19. The Generation of an Aerodynamical Propeller Using Partial Differential Equations -- 1. Introduction -- 2. The PDE Method -- 3. Generating the Propeller -- 4. Aerodynamics -- 5. Influence of Parameters -- References -- Chapter 20. Fast Computation of Cross-Validated Robust Splines and Other Non-linear Smoothing Splines -- 1. Introduction -- 2. Choice of p by Generalized Cross-validation -- 3. Monte-Carlo Computation of Trace Terms -- 4. Numerical Experiments -- References -- Chapter 21. Szasz-Mirakyan Quasi-interpolants -- 1. Introduction and Definitions -- 2. Norms of the Left Quasi-interpolants -- 3. Woronovskaya-type Relation -- Acknowledgements -- References -- Chapter 22. Statistical Check On The Smoothing Parameter of a Method for Inversion of Fourier Series -- References -- Chapter 23. A General Method of Treating Degenerate Bézier Patches -- 1. Introduction -- 2. Computing Geometric Features -- 3. Geometric Continuity Constraints over Degenerate Patches -- References -- Chapter 24. G1 Smooth Connection Between Rectangular and Triangular Bézier Patches at a Common Corner -- 1. G1 Continuity Between Two Adjacent Bézier Patches -- 2. G1 Continuity Around a Mixed N-patch Corner -- References -- Chapter 25. Regularity Conditions for a Class of Geometrically Continuous Curves and Surfaces -- 1. Introduction -- 2. Regularity Conditions for Bell-Shaped Functions -- 3. Two Classes of Bell-Shaped Functions -- References -- Chapter 26. Splines and Digital Signal Processing -- 1. Introduction -- 2. B-Spline Digital Filters -- References -- Chapter 27. B-Rational Curves and Surfaces N-Rational Splines -- 1. Introduction -- 2. General Framework -- 3. The (BR) Curves -- 4. The N-rational Splines -- 5. The (SBR) Surfaces -- References -- Chapter 28. Reparametrizations of Polynomial and Rational Curves.
1. Introduction -- 2. Homographie Transformation -- 3. Rational Quadratic Transformation -- 4. Conclusion -- References -- Chapter 29. Numerical Stability of Geometric Algorithms -- References -- Chapter 30. Solving Implicit ODEs by Simplicial Methods -- 1. Introduction -- 2. PL Approximation of Implicit Manifolds -- 3. Implicit Differential Equations and Singularities of Mappings -- 4. A Simplicial Method to Solve Implicit ODEs -- References -- Chapter 31. On the Power of a posteriori Error Estimation for Numerical Integration and Function Approximation -- 1. Introduction -- 2. Proof of the Theorems -- References -- Chapter 32. Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two -- 1. Introduction -- 2. Multiresolution Analysis -- 3. Symbol Calculus -- 4. Stability -- 5. Linear Independence -- 6. Pre-Wavelet Decomposition -- 7. Wavelet Decomposition -- 8. Extensibility -- References -- Chapter 33. Elastica and Minimal-Energy Splines -- 1. Elastica -- 2. Minimal-energy Spline Segments -- 3. Minimal-energy Splines -- References -- Chapter 34. A Distributed Algorithm for Surface/Plane Intersection -- 1. Introduction -- 2. The Subdivision Step -- 3. The Intersection Step -- 4. Experimental Examples -- 5. Conclusions -- References -- Chapter 35. Construction of Exponential Tension B-splines of Arbitrary Order -- 1. B-splines -- 2. Conversion to a Bézier-like Form -- References -- Chapter 36. On the Almost Sure Limit of Probabilistic Recovery -- References -- Chapter 37. A New Curve Tracing Algorithm and Some Applications -- 1. Introduction -- 2. Curve Tracing Algorithm -- 3. Applications -- References -- Chapter 38. Pseudo-Cubic Weighted Splines Can Be C2 or G2 -- 1. Introduction -- 2. General Weighted Interpolating Spline -- 3. Interpolating q-spline of Order 2.
4. Smoothing q-spline of Order 2 -- 5. Polar Representation of q-splines -- References -- Chapter 39. Composite Cr-Triangular Finite Elements of PS Type on a Three Direction Mesh -- 1. Introduction -- 2. PS Triangles of Class C2s -- 3. PS Triangles of Class C2s+1 -- References -- Chapter 40. Dynamic Segmentation: Finding the Edge With Snake Splines -- 1. Introduction -- 2. Modelling Curves Or Surfaces With Spline Functions -- 3. Strength Fields and Potential Convolving -- 4. Adaptative Evolution of the Snake-Spline -- 5. Preliminary Results -- 6. Conclusion -- References -- Chapter 41. Recent Developments in the Strang-Fix Theory for Approximation Orders -- 1. Introduction -- 2. The Strang-Fix Theory -- 3. Functions with Rapid Decay -- References -- Chapter 42. Aligning Frames with the Tangent Curve of a B-spline Curve -- 1. Introduction -- 2. Linear Interpolation of Frames in 2D -- 3. 5-spline Approximation of Frames in 2D -- 4. Frame Splines in 3D -- References -- Chapter 43. Error Estimates for Interpolation by Generalized Splines -- 1. Introduction -- 2. Variational Formulation -- 3. Error Estimates -- 4. Miscellaneous Examples and Remarks -- References -- Chapter 44. Varying the Shape Parameters of Rational Continuity -- 1. Introduction -- 2. Rational Continuity -- 3. Basis Functions -- 4. Conclusion -- References -- Chapter 45. Detecting Cusps and Inflection Points in Curves -- 1. Introduction -- 2. Parametric Curves -- 3. Necessary Condition for Cusps -- 4. Necessary and Sufficient Condition for Cusps -- 5. Proper Parametrizations -- References -- Chapter 46. Image-like Surfaces: Parallel Least Squares Approximation Methods -- 1. Introduction -- 2. Parallel Least Squares Methods -- References -- Chapter 47. Local Kriging Interpolation: Application to Scattered Data on the Sphere -- 1. Introduction.
2. Local Kriging Interpolation or Approximation on the Sphere.
Summary: Curves and Surfaces provides information pertinent to the fundamental aspects of approximation theory with emphasis on approximation of images, surface compression, wavelets, and tomography. This book covers a variety of topics, including error estimates for multiquadratic interpolation, spline manifolds, and vector spline approximation. Organized into 77 chapters, this book begins with an overview of the method, based on a local Taylor expansion of the final curve, for computing the parameter values. This text then presents a vector approximation based on general spline function theory. Other chapters consider a nonparametric technique for estimating under random censorship the amplitude of a change point in change point hazard models. This book discusses as well the algorithm for ray tracing rational parametric surfaces based on inversion and implicitization. The final chapter deals with the results concerning the norm of the interpolation operator and error estimates for a square domain. This book is a valuable resource for mathematicians.
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Electronic Book UT Tyler Online
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QA221 -- .C87 1991 (Browse shelf) http://ebookcentral.proquest.com/lib/uttyler/detail.action?docID=1901710 Available EBC1901710

Front Cover -- Curves and Surfaces -- Copyright Page -- Table of Contents -- PREFACE -- CONTRIBUTORS -- Chapter 1. Parametrization for Data Approximation -- 1. Introduction -- 2. Local Taylor Expansion of the Curve -- 3. Criteria -- 4. Application to Approximation with B-Splines -- References -- Chapter 2. A Vector Spline Approximation With Application to Meteorology -- 1. Introduction -- 2. The Minimization Problem -- 3. Solution of P α,β -- 4. Limit Problems -- 5. Numerical Results -- References -- Chapter 3. Kernel Estimation in Change-Point Hazard Rate Models -- 1. Introduction -- 2. The Model -- 3. Main Results -- 4. Simulation Results and Concluding Remarks -- References -- Chapter 4. Spline Manifolds -- 1. Introduction -- 2. Basic Tools -- 3. A Fundamental Result -- 4. Spline Manifolds -- 5. P(D)-positive Vectorial Distributions -- 6. Extension of P(D)-spline -- References -- Chapter 5. Use of Simulated Annealing to Construct Triangular Facet Surfaces -- 1. Introduction -- 2. Optimal Triangulations -- 3. Locally Optimal Triangulations and Edge Swapping -- 4. Simulated Annealing -- 5. An Example -- 6. Conclusions -- References -- Chapter 6. G1 and G2 Continuity Between (SBR) Surfaces -- 1. Geometric Framework -- 2. Analytical Conditions -- 3. Geometric Continuity Between Two Rectangular (SBR) -- References -- Chapter 7. Ray Tracing Rational Parametric Surfaces -- 1. Introduction -- 2. Implicitization -- 3. Intersection Problem -- 4. Discussion -- References -- Chapter 8. Energy-Based Segmentation of Sparse Range Data -- Abstract -- 1. Segmentation: Introduction and Background -- 2. Definition of the Model of World Surfaces -- 3. Experimentation -- References -- Chapter 9. Error Estimates for Multiquadric Interpolation -- 1. Introduction and Statement of Main Result -- 2. Proof of Theorem 1 -- References.

Chapter 10. A Geometrical Analysis for a Data Compression of 3D Anatomical Structures -- 1. Introduction -- 2. Data Format -- 3. Feature Extraction -- 4. Data Compression -- 5. Contour Reconstruction -- 6. Conclusion -- References -- Chapter 11. Ck Continuity of (SBR) Surfaces -- 1. Framework -- 2. Rectangular (SBR) Surfaces -- 3. Triangular (SBR) Surfaces -- References -- Chapter 12. A Note on Piecewise Monotonie Bivariate Interpolation -- 1. Introduction -- 2. Outline of the Algorithm -- 3. Conclusions -- References -- Chapter 13. Real-Time Signal Analysis with Quasi-Interpolatory Splines and Wavelets -- 1. Introduction -- 2. Spline Sampling of Digital Signals -- 3. Wavelet Signal Decomposition -- 4. Wavelet Signal Reconstruction -- References -- Chapter 14. Polynomial Expansions for Cardinal Interpolants and Orthonormal Wavelets -- 1. Introduction -- 2. The Case of Real Φ -- 3. The Case of Complex Φ -- 4. Construction of Quasi-Interpolation Operators -- References -- Chapter 15. Realtime Pipelined Spline Data Fitting for Sketched Curves -- 1. Introduction -- 2. Background -- 3. Data Reduction -- 4. Pipelining the Algorithm -- 6. Remarks -- References -- Chapter 16. Remarks on Digital Terrain Modelling Accuracy -- 1. On Terrain Modelling Accuracy -- 2. The Displacement-Buckling Approach -- 3. A Graphical Approach -- References -- Chapter 17. Convexity and Bernstein-Bézier Polynomials -- 1. Introduction -- 2. Some Prerequisites -- 3. Planar Curves -- 4. Convexity of Functional Bézier Surfaces -- 5. Convexity of Piecewise Polynomial Surfaces -- 6. Convexity of Parametric Bézier Patches -- 7. Converse Theorems on Convexity -- References -- Chapter 18. How to Draw a Curve Using Geometrical Data -- 1. Introduction -- 2. Algebraic Regression -- 3. Automatic Ordering -- References.

Chapter 19. The Generation of an Aerodynamical Propeller Using Partial Differential Equations -- 1. Introduction -- 2. The PDE Method -- 3. Generating the Propeller -- 4. Aerodynamics -- 5. Influence of Parameters -- References -- Chapter 20. Fast Computation of Cross-Validated Robust Splines and Other Non-linear Smoothing Splines -- 1. Introduction -- 2. Choice of p by Generalized Cross-validation -- 3. Monte-Carlo Computation of Trace Terms -- 4. Numerical Experiments -- References -- Chapter 21. Szasz-Mirakyan Quasi-interpolants -- 1. Introduction and Definitions -- 2. Norms of the Left Quasi-interpolants -- 3. Woronovskaya-type Relation -- Acknowledgements -- References -- Chapter 22. Statistical Check On The Smoothing Parameter of a Method for Inversion of Fourier Series -- References -- Chapter 23. A General Method of Treating Degenerate Bézier Patches -- 1. Introduction -- 2. Computing Geometric Features -- 3. Geometric Continuity Constraints over Degenerate Patches -- References -- Chapter 24. G1 Smooth Connection Between Rectangular and Triangular Bézier Patches at a Common Corner -- 1. G1 Continuity Between Two Adjacent Bézier Patches -- 2. G1 Continuity Around a Mixed N-patch Corner -- References -- Chapter 25. Regularity Conditions for a Class of Geometrically Continuous Curves and Surfaces -- 1. Introduction -- 2. Regularity Conditions for Bell-Shaped Functions -- 3. Two Classes of Bell-Shaped Functions -- References -- Chapter 26. Splines and Digital Signal Processing -- 1. Introduction -- 2. B-Spline Digital Filters -- References -- Chapter 27. B-Rational Curves and Surfaces N-Rational Splines -- 1. Introduction -- 2. General Framework -- 3. The (BR) Curves -- 4. The N-rational Splines -- 5. The (SBR) Surfaces -- References -- Chapter 28. Reparametrizations of Polynomial and Rational Curves.

1. Introduction -- 2. Homographie Transformation -- 3. Rational Quadratic Transformation -- 4. Conclusion -- References -- Chapter 29. Numerical Stability of Geometric Algorithms -- References -- Chapter 30. Solving Implicit ODEs by Simplicial Methods -- 1. Introduction -- 2. PL Approximation of Implicit Manifolds -- 3. Implicit Differential Equations and Singularities of Mappings -- 4. A Simplicial Method to Solve Implicit ODEs -- References -- Chapter 31. On the Power of a posteriori Error Estimation for Numerical Integration and Function Approximation -- 1. Introduction -- 2. Proof of the Theorems -- References -- Chapter 32. Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two -- 1. Introduction -- 2. Multiresolution Analysis -- 3. Symbol Calculus -- 4. Stability -- 5. Linear Independence -- 6. Pre-Wavelet Decomposition -- 7. Wavelet Decomposition -- 8. Extensibility -- References -- Chapter 33. Elastica and Minimal-Energy Splines -- 1. Elastica -- 2. Minimal-energy Spline Segments -- 3. Minimal-energy Splines -- References -- Chapter 34. A Distributed Algorithm for Surface/Plane Intersection -- 1. Introduction -- 2. The Subdivision Step -- 3. The Intersection Step -- 4. Experimental Examples -- 5. Conclusions -- References -- Chapter 35. Construction of Exponential Tension B-splines of Arbitrary Order -- 1. B-splines -- 2. Conversion to a Bézier-like Form -- References -- Chapter 36. On the Almost Sure Limit of Probabilistic Recovery -- References -- Chapter 37. A New Curve Tracing Algorithm and Some Applications -- 1. Introduction -- 2. Curve Tracing Algorithm -- 3. Applications -- References -- Chapter 38. Pseudo-Cubic Weighted Splines Can Be C2 or G2 -- 1. Introduction -- 2. General Weighted Interpolating Spline -- 3. Interpolating q-spline of Order 2.

4. Smoothing q-spline of Order 2 -- 5. Polar Representation of q-splines -- References -- Chapter 39. Composite Cr-Triangular Finite Elements of PS Type on a Three Direction Mesh -- 1. Introduction -- 2. PS Triangles of Class C2s -- 3. PS Triangles of Class C2s+1 -- References -- Chapter 40. Dynamic Segmentation: Finding the Edge With Snake Splines -- 1. Introduction -- 2. Modelling Curves Or Surfaces With Spline Functions -- 3. Strength Fields and Potential Convolving -- 4. Adaptative Evolution of the Snake-Spline -- 5. Preliminary Results -- 6. Conclusion -- References -- Chapter 41. Recent Developments in the Strang-Fix Theory for Approximation Orders -- 1. Introduction -- 2. The Strang-Fix Theory -- 3. Functions with Rapid Decay -- References -- Chapter 42. Aligning Frames with the Tangent Curve of a B-spline Curve -- 1. Introduction -- 2. Linear Interpolation of Frames in 2D -- 3. 5-spline Approximation of Frames in 2D -- 4. Frame Splines in 3D -- References -- Chapter 43. Error Estimates for Interpolation by Generalized Splines -- 1. Introduction -- 2. Variational Formulation -- 3. Error Estimates -- 4. Miscellaneous Examples and Remarks -- References -- Chapter 44. Varying the Shape Parameters of Rational Continuity -- 1. Introduction -- 2. Rational Continuity -- 3. Basis Functions -- 4. Conclusion -- References -- Chapter 45. Detecting Cusps and Inflection Points in Curves -- 1. Introduction -- 2. Parametric Curves -- 3. Necessary Condition for Cusps -- 4. Necessary and Sufficient Condition for Cusps -- 5. Proper Parametrizations -- References -- Chapter 46. Image-like Surfaces: Parallel Least Squares Approximation Methods -- 1. Introduction -- 2. Parallel Least Squares Methods -- References -- Chapter 47. Local Kriging Interpolation: Application to Scattered Data on the Sphere -- 1. Introduction.

2. Local Kriging Interpolation or Approximation on the Sphere.

Curves and Surfaces provides information pertinent to the fundamental aspects of approximation theory with emphasis on approximation of images, surface compression, wavelets, and tomography. This book covers a variety of topics, including error estimates for multiquadratic interpolation, spline manifolds, and vector spline approximation. Organized into 77 chapters, this book begins with an overview of the method, based on a local Taylor expansion of the final curve, for computing the parameter values. This text then presents a vector approximation based on general spline function theory. Other chapters consider a nonparametric technique for estimating under random censorship the amplitude of a change point in change point hazard models. This book discusses as well the algorithm for ray tracing rational parametric surfaces based on inversion and implicitization. The final chapter deals with the results concerning the norm of the interpolation operator and error estimates for a square domain. This book is a valuable resource for mathematicians.

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