Colloquium

In this talk, I will present an application of algebraic geometry to geometric modeling. Geometric modeling uses curves and surfaces to represent objects on a computer. The computer models of geometric objects are widely used in industrial design and manufacture, and in architecture. There are two basic shapes for surfaces — triangular and rectangular Bézier patches. Recently, modelers have developed diverse multi-sided patches capable of producing more robust alternatives to the standard 3 and 4-sided patches. Among these, toric patches, based on the geometry of toric varieties, have gained particular attention.

The widespread adoption of Bézier patches is due in part to their possessing many useful mathematical properties. In this talk I will discuss some properties of these multisided toric Bézier patches. In particular, we will answer the following question. How does the control net controls the underlying Bézier surface patch. The case of Bézier curves is well-understood. The control polygon of a Bézier curve is well-defined and has geometric significance — there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bézier surface patch, there are many possible polyhedral control structures, and none are canonical. We propose a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain \(C^0\) spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bézier patch when the weights are allowed to vary.

Our results rely upon a construction in computational algebraic geometry called toric degeneration. This is ongoing work with Frank Sottile and Chungang Zhu.