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
of this talk.
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