Local control of surfaces generated using partial differential equations

Abstract We discuss methods of exercising local control over surfaces that are generated as solutions of partial differential equations (PDEs). By examining the way in which the choice of the PDE influences the form of the solution, we show how required properties of a surface may be realised by using a suitably chosen equation. Also, we discuss the way in which a B-spline surface may be grafted on to the PDE surface to achieve small scale changes.

[1]  P. E. Bezier,et al.  Example of an existing system in the motor industry: the Unisurf system , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  Malcolm I. G. Bloor,et al.  A strategy for the automated design of mechanical parts , 1993, Solid Modeling and Applications.

[3]  John C. Platt,et al.  Constraints methods for flexible models , 1988, SIGGRAPH.

[4]  Pierre Bezier,et al.  The Mathematical Basis of the Unisurf CAD System , 1986 .

[5]  Joshua Kiddy K. Asamoah,et al.  Fractal–fractional age-structure study of omicron SARS-CoV-2 variant transmission dynamics , 2022, Partial Differential Equations in Applied Mathematics.

[6]  I. Faux,et al.  Computational Geometry for Design and Manufacture , 1979 .

[7]  George William Celniker,et al.  ShapeWright--finite element based free-form shape design , 1990 .

[8]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[9]  G. Hedstrom,et al.  Numerical Solution of Partial Differential Equations , 1966 .

[10]  Christopher Williams,et al.  Use of structural analogy in generation of smooth surfaces for engineering purposes , 1987 .

[11]  B. Barsky,et al.  An Introduction to Splines for Use in Computer Graphics and Geometric Modeling , 1987 .

[12]  S. A. Coons.  SURFACES FOR COMPUTER-AIDED DESIGN OF SPACE FORMS , 1967 .

[13]  Demetri Terzopoulos,et al.  Physically based models with rigid and deformable components , 1988, IEEE Computer Graphics and Applications.

[14]  M. I. G. Bloor,et al.  Blend design as a boundary-value problem , 1989 .

[15]  L. Piegl,et al.  Curve and surface constructions using rational B-splines , 1987 .

[16]  Malcolm I. G. Bloor,et al.  Representing PDE surfaces in terms of B-splines , 1990, Comput. Aided Des..

[17]  Malcolm I. G. Bloor,et al.  Using partial differential equations to generate free-form surfaces , 1990, Comput. Aided Des..

[18]  Malcolm I. G. Bloor,et al.  Functionality in blend design , 1991, Comput. Aided Des..

[19]  Malcolm I. G. Bloor,et al.  An automated method for the incorporation of functionality in the geometric design of a shell , 1993, Solid Modeling and Applications.

[20]  S. A. Coons.  Modification of the shape of piecewise curves , 1977 .

[21]  Charles D. Woodward,et al.  Cross-sectional design of B-spline surfaces , 1987, Comput. Graph..

[22]  Michael J. Wilson,et al.  Generating blend surfaces using partial differential equations , 1989 .

[23]  Wayne Tiller,et al.  Rational B-Splines for Curve and Surface Representation , 1983, IEEE Computer Graphics and Applications.