Mathematical Methods in Computer Aided Geometric Design

Mathematical Methods in Computer Aided Geometric Design covers the proceedings of the 1988 International Conference by the same title, held at the University of Oslo, Norway. This text contains papers based on the survey lectures, along with 33 full-length research papers.

This book is composed of 39 chapters and begins with surveys of scattered data interpolation, spline elastic manifolds, geometry processing, the properties of Bézier curves, and Gröbner basis methods for multivariate splines. The next chapters deal with the principles of box splines, smooth piecewise quadric surfaces, some applications of hierarchical segmentations of algebraic curves, nonlinear parameters of splines, and algebraic aspects of geometric continuity. These topics are followed by discussions of shape preserving representations, box-spline surfaces, subdivision algorithm parallelization, interpolation systems, and the finite element method. Other chapters explore the concept and applications of uniform bivariate hermite interpolation, an algorithm for smooth interpolation, and the three B-spline constructions. The concluding chapters consider the three B-spline constructions, design tools for shaping spline models, approximation of surfaces constrained by a differential equation, and a general subdivision theorem for Bézier triangles.

This book will prove useful to mathematicians and advance mathematics students.

Contenu

Preface
Participants
Scattered Data Interpolation in Three or More Variables
Some Applications of Discrete Dm Splines
Spline Elastic Manifolds
Geometry Processing: Curvature Analysis and Surface-Surface Intersection
Three Examples of Dual Properties of Bézier Curves
What is the Natural Generalization of a Bézier Curve?
Convexity and a Multidimensional Version of the Variation Diminishing Property of Bernstein Polynomials
Gröbner Basis Methods for Multivariate Splines
On Finite Element Interpolation Problems
The Design of Curves and Surfaces by Subdivision Algorithms
A Data Dependent Parametrization for Spline Approximation
On the Evaluation of Box Splines
Smooth Piecewise Quadric Surfaces
Inserting New Knots Into Beta-Spline Curves
Recursive Subdivision and Iteration in Intersections and Related Problems
Rational Curves and Surfaces
Hierarchical Segmentations of Algebraic Curves and Some Applications
An Algorithm for Shape Preserving Parametric Interpolating Curves with G2 Continuity
Knot Selection for Parametric Spline Interpolation
Splines and Estimation of Nonlinear Parameters
On Beta-Continuous Functions and Their Application to the Construction of Geometrically Continuous Curves and Surfaces
Algebraic Aspects of Geometric Continuity
Shape Preserving Representations
Geometric Continuity
Curvature Continuous Triangular Interpolants
Box-Spline Surfaces
Parallelization of the Subdivision Algorithm for Intersection of Bézier Curves on the FPS T20
Composite Quadrilateral Finite Elements of Class Cr
A Knot Removal Strategy for Scattered Data in R2
Interpolation Systems and the Finite Element Method
Uniform Bivariate Hermite Interpolation
A Survey of Applications of an Affine Invariant Norm
An Algorithm for Smooth Interpolation to Scattered Data in R2
Some Remarks on Three B-Spline Constructions
Modified B-Spline Approximation for Quasi-Interpolation or Filtering
Design Tools for Shaping Spline Models
A Process Oriented Design Method for Three-Dimensional CAD Systems
Open Questions in the Application of Multivariate B-Splines
On Global GC2 Convexity Preserving Interpolation of Planar Curves by Piecewise Bézier Polynomials
Best Interpolation with Free Nodes by Closed Curves
Segmentation Operators on Coons' Patches
A General Subdivision Theorem for Bézier Triangles
Cardinal Interpolation with Translates of Shifted Bivariate Box-Splines
Approximation of Surfaces Constrained by a Differential Equation Using Simplex Splines
A Construction for VC1 Continuity of Rational Bézier Patches