Rational Conchoids of Algebraic Curves

TL;DR

This paper investigates the conditions under which conchoids of algebraic curves are rational, providing algorithms for analysis and parametrization, and applying results to classical curves like conics and limaçons.

Contribution

It establishes that only rational base curves generate rational conchoids, introduces an algorithm for rationality analysis, and parametrizes conchoids of notable curves such as conics and limaçons.

Findings

Conchoids of rational curves are rational.

The rationality of a conchoid depends on the base curve and focus, not on the distance.

The paper provides explicit parametrizations for conchoids of conics and limaçons.

Abstract

We study the rationality of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the isotropic lines, of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. Also, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the {\it Lima\c{c}ons of Pascal}. We also parametrize the conchoids of {\it Nicomedes}. Finally, we show to find the focuses from where the conchoid is rational or with two rational components.