Analytic Functions in Geometric Modeling

tation for polynomials in one or two variables. Nevertheless, the
number of free parameters is limited by the degree of the polyno-
mial. B-splines provide an alternative to Bezier schemes, intro-
ducing more parameters and keeping the degree low while preserv-
ing the intuitive interpretation of Bezier curves and surfaces.
But B-splines cannot represent even simple rational functions ex-
actly. More general yet, NURBS provide a new model, capable of
exactly reproducing arcs of circles. However, the parameteriza-
tion of these circles is highly non-uniform.

Analytic functions are a very rich family that includes polynomial
and trigonometric functions as well as many other standard
functions that can be used for the representation of curves and
surfaces. In this talk, we present a framework for analytic functions
adapted to geometric modeling. Poisson curves and surfaces are to
analytic functions what Bezier curves are to polynomials. Algo-
rithms and tools like subdivision, Marsden's identity and blos-
soming are extended from the polynomial to the analytic setting.
Inherently in this model, convergence issues arise. We will pre-
sent results on convergence that prove our method and model
sound, as well as underlying algorithms that have a practical
use.