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MiS Preprint

Measuring the local non-convexity of real algebraic curves

Miruna-Ştefana Sorea


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 Poincare-Reeb graph, whose role is to encode the shape of such curves and to allow us to quantify their non-convexity. Moreover, we prove that in this setting the Poincare-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.

Jul 22, 2019
Jul 24, 2019
MSC Codes:
14P25, 14P05, 14H20, 14B05, 05C05, 14Q05, 26C, 58K
strict local minimum, Poincare-Reeb tree, non-convexity, level curve, stabilisation, real algebraic curve, polar curve, star domain

Related publications

2022 Repository Open Access
Miruna-Stefana Sorea

Measuring the local non-convexity of real algebraic curves

In: Journal of symbolic computation, 109 (2022), pp. 482-509