Shape determination for real curves and surfaces

In this paper we describe algorithms to find the shape of a real algebraic curve in P2 and the topology of a real algebraic surface in P3. The algorithm runs as follows: choose a point O∈P2 outside the curve C and in general position with respect to C; consider the projection π from the curve to P1 with center 0; determine the critical points and the critical values of π, and the inverse image of the critical values. Investigating the mutual position of these points, we obtain two finite sequences of integers, from which we obtain explicitly the shape of C, as a finite set corresponding to the set of branches of C, and a partial order relation on this set, corresponding to the inclusion between branches. The algorithm for surfaces considers the variation of the shape of the curve in a pencil, to find the number of connected components and their rational homology; this describes completly the topology of the surface. We discuss some results on the explicit computer implementation of the algorithm for curves, which has proved to be rapid and reliable