Numerical proper reparametrization of parametric plane curves

Characterizing the finiteness of the Hausdorff distance between two algebraic curves

“…[7,[14][15][16][17]20]). More precisely, in approximate parametrization problems, the Hausdorff distance is an essential tool for measuring the resemblance between the input and the output curves (which could not be bounded).…”

“…This parametrization is ''almost'' improper (see Section 3 in [17]). The algorithm presented in [17] returns the curve C defined by the parametrization P (s) = (p 1 (s)/q 1 (s), p 2 (s)/q 2 (s)), where p 1 (s) = −60688159524533550201(20s 2 − 2s − 1), q 1 (s) = 60688159524533550201s 3 − 18449333330658180s 2 + 60081278530101s − 265814138756, p 2 (s) = 10975164641(1105s 2 + 1104s − 368), q 2 (s) = 92(21953640540s 2 + 21950329282s − 21975135).…”

“…In particular, we show how the Hausdorff distance between two algebraic curves obtained by applying approximate parametrization problems is finite (we consider examples presented in [7,8,17]). Once one has ensured that the Hausdorff distance between the two curves is finite, some additional existing methods can be applied to estimate the Hausdorff distance between two not bounded curves (see [8]), or for a chosen bounded frame of the input curves (see e.g.…”

“…In [17] given a rational plane curve, C, defined by a parametrization P that is ''almost'' improper (numerically speaking), a method for computing a new rational plane curve, C, defined by a proper parametrization, P , is presented. In [17], some bounds for measuring the closeness between C and C are presented but it can be applied for bounded frames of the curves (see Section 4 in [17]). In order to have a total analysis of both curves and to ensure the effectiveness of the method presented, the behavior at infinity has to be studied, and the finiteness of the Hausdorff distance between C and C has to be guaranteed.…”

“…[7,[14][15][16][17]). In that problem, given an affine curve C (say that it is a perturbation of a rational curve), the goal is to compute a rational proper parametrization of a rational affine curve C near C (one may state the problem also for surfaces; see [18]).…”