Peter Alfeld

Scattered data interpolation in three or more variables

This is a survey of techniques for the interpolation of scattered data in three or more independent variables. It covers schemes that can be used for any number of variables as well as schemes specifically designed for three variables. Emphasis is on breadth rather than depth, but there are explicit illustrations of different techniques used in the solution of multivariate interpolation problems.

Fitting scattered data on sphere-like surfaces using spherical splines

Abstract Spaces of polynomial splines defined on planar traingulations are very useful tools for fitting scattered data in the plane. Recently, [4, 5], using homogeneous polynomials, we have developed analogous spline spaces defined on triangulations on the sphere and on sphere-like surfaces. Using these spaces, it is possible to construct analogs of many of the classical interpolation and fitting methods. Here we examine some of the more interesting ones is detail. For interpolation, we discuss macro-element and minimal energy splines, and for fitting, we consider discrete least squares and penalized least squares.

The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1

We consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessingr continuous derivatives globally. Morgan and Scott constructed a basis in the case wherer=1 andd≥5. The purpose of this paper is to extend the dimension part of their result tor≥0 andd≥4r+l. We use Bézier nets as a crucial tool in deriving the dimension of such spaces.

An explicit basis for C 1 quartic by various bivariate splines

We establish the dimension of the space of

A trivariate clough-tocher scheme for tetrahedral data

C^1

Bernstein-Be´zier polynomials on spheres and sphere-like surfaces

bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain. Our approach is to construct a minimal determining set and an associated explicit basis for the space. For general triangulations, the minimal determining set must be defined globally.

On the dimension of bivariate spline spaces of smoothnessr and degreed=3r+1

An interpolation scheme is described for values of position, gradient and Hessian at scattered points in three variables. The domain is assumed to have been tesselated into tetrahedra. The interpolant has local support, is globally once differentiable, piecewise polynomial, and reproduces polynomials of degree up to three exactly. The scheme has been implemented in a FORTRAN research code.

A bivariate C2 Clough-Tocher scheme

Abstract In this paper we discuss a natural way to define barycentric coordinates on general sphere-like surfaces. This leads to a theory of Bernstein-Bezier polynomials which parallels the familiar planar case. Our constructions are based on a study of homogeneous polynomials on trihedra in R 3. The special case of Bernstein-Bezier polynomials on a sphere is considered in detail.

Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d ≥ r + 1

SummaryWe consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend≧3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr≧1 andd=3r+1 for almost all triangulations.

Smooth Macro-Elements Based on Powell–Sabin Triangle Splits

A Clough-Tocher like interpolation scheme is described for values of position, gradient and hessian at scattered points in two variables. The domain is assumed to have been triangulated. The interpolant has local support, is globally twice differentiable, piecewise polynomial, and reproduces polynomials of degree up to three exactly.