Shape of Curves and Surfaces: the Combinatorics

This paper develops some points that were only sketched in the companion paper [6], in particular explaining the combinatorial part of the algorithms for the computation of the topology of real plane curves and space surfaces. The origin of these algorithms is [11].In particular, for the curves, we compute the shape through a computation of the intersection of the curve with the vertical tangents. From the number and placement of the roots of the intersection we can derive an explicit description of the shape. For the surfaces, we derive the information on the topology considering the change of the shape of a plane section, through a computation of the rank of the homology groups of the connected components.