  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.
