Parametrization of Algebraic Curves over Optimal Field Extensions

Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation

et al. 2000

Self Cite

A canal surface S, generated by a parametrized curve m(t), in R 3 is the envelope of the set of spheres with radius r(t) centered at m(t). This concept generalizes the classical offsets (for r(t) = const) of plane curves. In this paper we develop elementary symbolic methods for generating a rational parametrization of canal surfaces generated by rational curves m(t) with rational radius variation r(t). In a pipe surface r(t) is constant.

Section: Introductionmentioning

confidence: 99%

Symbolic parametrization of pipe and canal surfaces

Landsmann

Schicho

Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation

et al. 2000

Self Cite

“…In the next proposition some special cases of rational linear subsystem are analyzed. The following Proposition 1 and Theorem 2 are proved in [SW97].…”

Section: A Real Parametrization Algorithmmentioning

confidence: 93%

Real parametrization of algebraic curves

Sendra

Artificial Intelligence and Symbolic Computation

1998

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There are various algorithms known for deciding the parametrizability (rationality) of a plane algebraic curve, and if the curve is rational, actually computing a parametrization. Optimality criteria such as low degrees in the parametrization or low degree field extensions are met by some parametrization algorithms. In this paper we investigate real curves. Given a parametrizable plane curve over the complex numbers, we decide whether it is in fact real. Furthermore, we discuss methods for actually computing a real parametrization for a parametrizable real curve.

“…4 The question when a rational algebraic plane curve over Q is parametrizable over R is treated in section 3.3 ("Parametrizing over the reals") of [12]. We state here the main result.…”

Section: Algorithm For Solving the Legendre Equationmentioning

confidence: 99%

“…We state here the main result. Theorem 3.2 (in [12]) A rational algebraic plane curve over Q is parametrizable over R if and only if it is not birationally equivalent over R to the conic x 2 + y 2 + z 2 .…”

Section: Algorithm For Solving the Legendre Equationmentioning

confidence: 99%

“…The construction of such an "optimal" parametrization requires O(deg(f )) birational transformations. Every one of these transformations decreases the degree of the curve by 2, and thus ultimately leads to either a line (in the odd degree case) or to an irreducible conic C 2 defined over the same field as the original curve C. This reduction process was abbreviated in [12] and actually the corresponding conic can simply be interpolated. Every point on the conic C 2 corresponds rationally to a point on C and vice versa (with finitely many exceptions).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Points on algebraic curves and the parametrization problem

Hillgarter

Automated Deduction in Geometry

1997

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Abstract.A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is rationally parametrizable and if so computing such a parametrization. In all these approaches we ultimately need some simple points on the curve. The field in which we can find such points crucially influences the coefficients in the resulting parametrization. We show how to find simple points over some practically interesting fields. Consequently, we are able to decide whether an algebraic curve defined over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric reasoning.