Larry L. Schumaker

Mathematical methods for curves and surfaces

We will deal with the translation surfaces which are the shapes generated by translating one curve along another one. We focus on the geometry of translation surfaces generated by two algebraic curves in space and study their properties, especially those useful for geometric modelling purposes. It is a classical result that each minimal surface may be obtained as a translation surface generated by an isotropic curve and its complex conjugate. Thus, we can study the minimal surfaces as special instances of translation surfaces. All the results about translation surfaces will be directly applied also to minimal surfaces. Finally, we present a construction of rational isotropic curves with a prescribed tangent field which leads to the description of all rational minimal surfaces. A close relation to surfaces with Pythagorean normals will be also discussed.

Local Spline Approximation Methods

Abstract : The construction of explicit polynomial spline approximation operators for real-valued functions defined on intervals or on reasonably behaved sets in higher dimensions is studied. The operators take the form Qf = the summation of lambda sub i f N sub i, where the N sub i are B-splines and the lambda sub i are appropriate linear functionals. Explicit operators are found which apply to wide classes of functions including continuous or integrable functions. Moreover, the operators are local and approximate smooth functions with an accuracy comparable to best spline approximation. They can be constructed without matrix inversion, local as well as global error bounds are obtained, and some of the error bounds are free of mesh restrictions. (Author)

Recent advances in wavelet analysis

This book covers recent advances in wavelet analysis and applications in areas including wavelets on bounded intervals, wavelet decomposition of special interest to statisticians, wavelets approach to differential and integral equations, analysis of subdivision operators, and wavelets related to problems in engineering and physics.

On the Dimension of Spaces Of Piecewise Polynomials in Two Variables

The purpose of this paper is to discuss the dimension of linear spaces of piecewise polynomials defined on triangulations of the plane. Such spaces are of considerable interest in general approximation and data fitting, as well as in the numerical solution of boundary-value problems by the finite-element method. We begin by defining the notion of triangulation.

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.

On the approximation power of bivariate splines

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces Sdr(Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain.

An explicit basis for C 1 quartic by various bivariate splines

We establish the dimension of the space of

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

C^1

Wavelets, images, and surface fitting

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.