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Abstract:
Geometric modeling builds computer models for industrial design and manufacture
from basic units, called patches. Many patches, including Bézier curves and
surfaces, are pieces of toric varieties, which are objects from algebraic
geometry. In 2002, Krasauskas generalized the standard Bézier patches to
multi-sided toric patches. While these offer the promise of greater design
flexibility, it is not clear whether they possess the desirable properties of
the standard patches. I will discuss work with Frank Sottile on elucidating some
geometric properties of toric patches such as linear precision, which is the
ability to replicate a linear function, self-intersection, and some global
deformations of a patch. I will start with a gentle introduction to geometric
modeling, Bézier patches and toric patches. The highlights of this talk are
(1) there are only three different toric surface patches with linear precision
(Bézier triangles, Bézier rectangles, and a new trapezoidal patch) (2) There
is an easy-to verify condition on a set of control points which implies that the
resulting patch has no self-intersection, for any choice of weights (3) a toric
curve can be deformed to its control polygon via a toric deformation (4) for
surfaces the previous statement is true only if the control polytope comes from
a regular triangulation.
This talk assumes knowledge of Calculus I and some
familiarity with polynomial algebra.
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