Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Abstract

In this paper we study situations when non-rational parameterizations of planar or space curves as results of certain geometric operations or constructions are obtained, in general. We focus especially on such cases in which one can identify a rational mapping which is a double cover of a rational curve. Hence, we deal with rational, elliptic or hyperelliptic curves that are birational to plane curves in the Weierstrass form and thus they are square-root parameterizable. We design a simple algorithm for computing an approximate (piecewise) rational parametrization using topological graphs of the Weierstrass curves. Predictable shapes reflecting a number of real roots of a univariate polynomial and a possibility to approximate easily the branches separately play a crucial role in the approximation algorithm. Our goal is not to give a comprehensive list of all such operations but to present at least selected interesting cases originated in geometric modelling and to show a unifying feature of the formulated method. We demonstrate our algorithm on a number of examples.

Computing the topology of a plane or space hyperelliptic curve

“…With some generality (see for instance [7]), we say that a curve C is hyperelliptic if there exists a generically two-to-one map C → R. Furthermore, such a curve (see for instance [25]) is birationally equivalent to a planar curve…”

“…Also, Eq. ( 3) is called the Weierstrass form of C. Notice (see p. 59 of [7]) that we can always transform the expression Eq.…”

“…Furthermore, this type of curves appears frequently in Computer Aided Geometric Design. A good account of the occurrence of hyperelliptic curves in this field is given in [7], where the problem of approximating hyperelliptic curves by means of rational parametrizations is addressed. As a brief summary of [7], non-rational offsets of rational planar curves and some bisector curves (line/rational curve, or circle/rational curve) are planar hyperelliptic curves.…”

“…A good account of the occurrence of hyperelliptic curves in this field is given in [7], where the problem of approximating hyperelliptic curves by means of rational parametrizations is addressed. As a brief summary of [7], non-rational offsets of rational planar curves and some bisector curves (line/rational curve, or circle/rational curve) are planar hyperelliptic curves. Contour curves of canal surfaces, intersections of two quadrics or intersections of a quadric and a ruled surface are examples of hyperelliptic curves in 3-space.…”

“…The method exploits similar ideas to [3], but focuses on offset curves, which have special properties. Finally, in [7] the problem of approximating a hyperelliptic curve by means of rational curves is considered. The Weierstrass form is also used in [7], but the goal is different, and in particular the computation of the topology of the hyperelliptic curve is not addressed.…”