Characterizing the finiteness of the Hausdorff distance between two algebraic curves

Abstract

a b s t r a c tIn this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the n-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 C and C is finite if and only if for each infinity branch of C there exists an infinity branch of C such that the terms with positive exponent in the corresponding series are the same, and reciprocally.

Recognizing projections of algebraic curves

“…One can find algorithms to approximate the Hausdorff distance in [4], [9], [16], [14], for the case of planar curves; in [17], [18], for space curves; and in [3], [5], [7] for surfaces. Additionally, some theoretical aspects of the question are treated in [6], for algebraic curves in n-space.…”