Mathematical Methods for Curves and Surfaces II

Morten Dæhlen, Tom Lyche, and Larry L. Schumaker, eds.

1998, Vanderbilt University Press, Nashville, TN

ISBN 0-8265-1315-8

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . x

A Practical Classification Method for Rational Quadratic Bézier Triangles with Respect to Quadrics

Gudrun Albrecht . . . . . . . . . . . . . . . . . . . . . . . 1

An Efficient and Robust Algorithm for Solving the Foot Point Problem I. J. Anderson, M. G. Cox, A. B. Forbes, J. C. Mason, and

D. A. Turner . . . . . . . . . . . . . . . . . . . . . . . . 9

Smooth Rational Cubic Spline Interpolation Christoph Baumgarten and Gerald Farin . . . . . . . . . . . . . 17

Generalized Fréchet Distance Between Curves Przemyslaw Bogacki and Stanley E. Weinstein . . . . . . . . . . 25

Elastic Splines with Tension Control Guido Brunnett and Jörg Wendt . . . . . . . . . . . . . . . . 33

Sign Consistency and Shape Properties J. M. Carnicer . . . . . . . . . . . . . . . . . . . . . . . . 41

Actuarial Science and Variational Splines R. Champion, C. T. Lenard and T. M. Mills . . . . . . . . . . . 49

Shape-Preserving Quasi-Interpolating Univariate Cubic Splines C. Conti, R. Morandi, C. Rabut . . . . . . . . . . . . . . . . 55

A Technique for the Construction of Generalized B-splines Costanza Conti and Christophe Rabut . . . . . . . . . . . . . . 63

Locally Linearly Independent Basis for C^{1} Bivariate Splines of Degree q ≥ 5

Oleg Davydov . . . . . . . . . . . . . . . . . . . . . . . . 71

Fair Surface Reconstruction from Point Clouds Ulrich Dietz . . . . . . . . . . . . . . . . . . . . . . . . . 79

Approximation with Rational B-spline Curves and Surfaces Bernhard Elsässer . . . . . . . . . . . . . . . . . . . . . . . 87 Cycles upon Cycles: An Anecdotal History

Rida T. Farouki and Joanne Rampersad . . . . . . . . . . . . . 95

Scattered Data Fitting on the Sphere Gregory E. Fasshauer and Larry L. Schumaker . . . . . . . . . . 117

On Spline Interpolation of Space Data Yu Yu Feng and Jernej Kozak . . . . . . . . . . . . . . . . . 167

The Smooth G^{2} Filling of a Rectangular Patch Hole Joining Four Heterogeneous Surfaces

J-C. Fiorot and O. Gibaru . . . . . . . . . . . . . . . . . . 175

Local and Global Convexity Preservation Michael S. Floater . . . . . . . . . . . . . . . . . . . . . . 183

Convergence of Cascade Algorithms T. N. T. Goodman and S. L. Lee . . . . . . . . . . . . . . . . 191

Refinable Vectors of Spline Functions T. N. T. Goodman and S. L. Lee . . . . . . . . . . . . . . . . 213

de Casteljau’s Algorithm Revisited Jens Gravesen . . . . . . . . . . . . . . . . . . . . . . . . 221

Triangulations from Repeated Bisection Bernd Hamann and Benjamin W. Jordan . . . . . . . . . . . . 229

The Normal Form of a Planar Curve And Its Application to Curve Design

Erich Hartmann . . . . . . . . . . . . . . . . . . . . . . . 237

Finite Element Thin Plate Splines for Data Mining Applications Markus Hegland, Stephen Roberts, and Irfan Altas . . . . . . . . 245

A Geometric Concept of Reverse Engineering of Shape: Approximation and Feature Lines

Josef Hoschek, Ulrich Dietz, and Wilhelm Wilke . . . . . . . . . 253

Convex Surface Fitting with Parametric Bézier Surfaces Bert Jüttler . . . . . . . . . . . . . . . . . . . . . . . . . 263

A Natural Geometric Norm for Multi-Channel Image Processing Ron Kimmel . . . . . . . . . . . . . . . . . . . . . . . . . 271

Fairing by Finite Difference Methods Leif Kobbelt . . . . . . . . . . . . . . . . . . . . . . . . . 279

Towards Free-form Fractal Modelling Ljubiša M. Kocić and Alba Chiara Simoncelli . . . . . . . . . . . 287

Fairness Approximation by Modified Discrete Smoothing D^{m}-splines A. Kouibia, M. Pasadas, and J. J. Torrens . . . . . . . . . . . . 295

Interpolating Curve with B-spline Curvature Function Mitsuru Kuroda, Masatake Higashi, Tsuyoshi Saitoh,

Yumiko Watanabe, and Tetsuzo Kuragano . . . . . . . . . . . 303

Some Families of Triangular Finite Elements Which Can Provide Sequences of Nested Spaces

A. Le Méhauté . . . . . . . . . . . . . . . . . . . . . . . 311

Computing Natural Division Lines on Free-Form Surfaces Based on Measured Data

Gábor Lukács and László Andor . . . . . . . . . . . . . . . . 319

Evaluation and Subdivision Algorithms for Totally Positive Systems E. Mainar and J. M. Peña . . . . . . . . . . . . . . . . . . . 327

A Parametric Hybrid Triangular Bézier Patch Stephen Mann and Matthew Davidchuk . . . . . . . . . . . . . 335

Comonotone Parametric Hermite Interpolation C. Manni and M. L. Sampoli . . . . . . . . . . . . . . . . . . 343

”Discrete” G^{1} Assembly of Patches Over Irregular Meshes T. Matskewich, O. Volpin, and M. Bercovier . . . . . . . . . . . 351

Joining Cyclide Patches along Quartic Boundary Curves Christoph Mäurer and Rimvydas Krasauskas . . . . . . . . . . . 359

Canal Surfaces and Inversive Geometry Marco Paluszny and Katja Bühler . . . . . . . . . . . . . . . . 367

From Wavelets to Multiwavelets Gerlind Plonka and Vasily Strela . . . . . . . . . . . . . . . . 375

An Analysis of Biarc Algorithms for Approximation Janet F. Poliakoff, Yew Kee Wong, and Peter D. Thomas . . . . . 401

Two-sided Patches Suitable for Inclusion in a B-Spline Surface Malcolm Sabin . . . . . . . . . . . . . . . . . . . . . . . . 409

Optimal Geometric Hermite Interpolation of Curves Robert Schaback . . . . . . . . . . . . . . . . . . . . . . . 417

An Interesting Class of Polynomial Vector Fields Gerik Scheuermann, Hans Hagen, and Heinz Krüger . . . . . . . . 429

Interpolation with Developable Strip–Surfaces Consisting of Cylinders and Cones

Mike Schneider . . . . . . . . . . . . . . . . . . . . . . . 437

Global Illumination Computations with Hierarchical Finite Element Methods

Hans-Peter Seidel, Philipp Slusallek, and Marc Stamminger …. 445

Boundary Curves in Reverse Engineering Vibeke Skytt . . . . . . . . . . . . . . . . . . . . . . . . . 461 Representation of Images by Surfaces and Higher Dimensional Manifolds in Non-Euclidean Space

Nir Sochen and Yehoshua Y. Zeevi . . . . . . . . . . . . . . . 469

Discrete Smoothing D^{m}-splines: Applications to Surface Fitting J. J. Torrens . . . . . . . . . . . . . . . . . . . . . . . . . 477

Spherical Pythagorean-Hodograph Curves Kenji Ueda . . . . . . . . . . . . . . . . . . . . . . . . . . 485

On Interactive Design Using The PDE Method Hassan Ugail, Malcolm I. G. Bloor, and Michael J. Wilson . . . . . 493

Vertex Blending: Problems and Solutions Tamás Várady and Chris M. Hoffmann . . . . . . . . . . . . . 501

Adding Constraints to Gordonesque Surfaces Marshall Walker . . . . . . . . . . . . . . . . . . . . . . . 529

Generalized Multiresolution Analysis for Arc Splines Johannes Wallner . . . . . . . . . . . . . . . . . . . . . . . 537

Analysis and Visualization of Geometric Curve Properties Geir Westgaard and Justus Heimann . . . . . . . . . . . . . . 545

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553