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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.
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