Peter Alfeld, --- Department of Mathematics, --- College of Science --- University of Utah

Two Talks given at Georgia Institute of Technology, December 4-5, 1997.

1. Bivariate Splines and the Bernstein Bézier form of a polynomial.


Multivariate splines are smooth piecewise polynomial functions defined on a tessellation of an underlying domain. They are used for interpolation and approximation of functions and data, the design of surfaces, and for the numerical solution of differential equations. Their basic properties are very simple in the univariate case (of one independent variable) and very complicated in the multivariate case. The Bernstein-Bézier form of a polynomial is a way of representing a polynomial that allows to approach algebraic problems (like how to ensure a differentiable transition from one polynomial piece to another) in geometric terms. This talk is focussed on bivariate splines defined on triangulations of a two-dimensional domain. It will summarize the history of bivariate splines, introduce the Bernstein-Bezier form, illustrate its use, and state some unsolved and apparently very difficult problems.

Multivariate Splines And The Four Color Map Problem.


This talk describes an approach to solving a particular multivariate spline problem using the same techniques that were used to solve the four color map problem. Thus we construct an unavoidable set of sub-triangulations using a discharging technique. The ideas are illustrated by proving a simpler result by the four color map techniques. That result, however, can also be obtained by simpler means. The work described here is very tentative but it does illustrate a perhaps unexpected connection between multivariate splines and the four color map problem. Understanding the talk requires familiarity with the Bernstein-Bezier form of a bivariate polynomial which is introduced in the preceding colloquium.

You can view annotated slides of this talk.


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