Parametrization of approximate algebraic surfaces by lines

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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 ( ɛ 1 / ( 2 d ) ) .