Annali dell’Università di Ferrara

, Volume 29, Issue 1, pp 87–109

# Shape determination for real curves and surfaces

• Patrizia Gianni
• Carlo Traverso
Article

## Summary

In this paper we describe algorithms to find the shape of a real algebraic curve inP 2 and the topology of a real algebraic surface inP 3.

The algorithm runs as follows: choose a pointOP 2 outside the curveC and in general position with respect toC; consider the projection π from the curve toP 1 with center0; 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 ofC, as a finite set corresponding to the set of branches ofC, 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.

A preliminary version of the algorithm for curves, with sketchy proofs, and containing a listing of theFortran programs, is contained in [G-T].

## Keywords

Boundary Component Open Disk Multiple Root Double Root Adjacency Relation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Riassunto

Si descrivono algoritmi per determinare la forma di una curva algebrica reale piana, e la topologia di una superficie reale inP 3.

La forma di una curvaC si ottiene proiettandoC suP 1, ed esaminando la posizione relativa dei punti critici rispetto alle immagini inverse dei valori critici. La forma di una superficieS si ottiene esaminando la variazione della forma in un fascio di sezioni piane, e ricavandone l'omologia razionale delle componenti connesse diS.

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