MEASURING THE LOCAL NON-CONVEXITY OF REAL ALGEBRAIC CURVES
Abstract
The goal of this paper is to measure the non-convexity of compact and smooth connected
components of real algebraic plane curves. We study these curves first in a general setting
and then in an asymptotic one. In particular, we consider sufficiently small levels of a real
bivariate polynomial in a small enough neighbourhood of a strict local minimum at the origin
of the real affine plane. We introduce and describe a new combinatorial object, called the
Poincaré-Reeb graph, whose role is to encode the shape of such curves and allow us to quantify
their non-convexity. Moreover, we prove that in this setting the Poincaré-Reeb graph is a
plane tree and can be used as a tool to study the asymptotic behaviour of level curves near
a strict local minimum. Finally, using the real polar curve, we show that locally the shape
of the levels stabilises and that no spiralling phenomena occur near the origin.
Origin : Files produced by the author(s)
Loading...