Characterizing the finiteness of the Hausdorff distance between two algebraic curves

In this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the nn-dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves CC and C¯ is finite if and only if for each infinity branch of CC there exists an infinity branch of C¯ such that the terms with positive exponent in the corresponding series are the same, and reciprocally.