Parametrization of approximate algebraic surfaces by lines

In this paper we present an algorithm for parametrizing approximate algebraic surfaces by lines. The algorithm is applicable to ɛ-irreducible algebraic surfaces of degree d having an ɛ-singularity of multiplicity d−1, and therefore it generalizes the existing approximate parametrization algorithms. In particular, given a tolerance ɛ>0 and an ɛ-irreducible algebraic surface V of degree d, the algorithm computes a new algebraic surface V¯, that is rational, as well as a rational parametrization of V¯. In addition, in the error analysis we show that the output surface V¯ and the input surface V are close. More precisely, we prove that V¯ lies in the offset region of V at distance, at most, O(ɛ1/(2d)).