We present an algorithm to compute the topology of a non-singular real algebraic surface S in RP^3, that is the number of its connected components and a topological model for each of them. Our strategy consists in computing the Euler characteristic of each connected component by means of a Morse-type investigation of S or of a suitably constructed compact affine surface. This procedure can be used to determine the topological type of an arbitrary non-singular surface; in particular it extends an existing algorithm applicable only to surfaces disjoint from a line.
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