On the Shape of Curves that are Rational in Polar Coordinates

TL;DR

This paper investigates the geometric properties of rational curves in polar coordinates, providing theoretical insights and an algorithm for visualizing their main features, with implementation in Maple.

Contribution

It offers a theoretical characterization of rational polar curves and develops an algorithm for plotting their key geometrical features, enhancing existing visualization tools.

Findings

Rational polar curves are generally non-algebraic in Cartesian coordinates.

The paper characterizes conditions for infinite self-intersections.

It introduces an algorithm implemented in Maple for visualizing these curves.

Abstract

In this paper we provide a computational approach to the shape of curves which are rational in polar coordinates, i.e. which are defined by means of a parametrization (r(t),\theta(t)) where both r(t),\theta(t) are rational functions. Our study includes theoretical aspects on the shape of these curves, and algorithmic results which eventually lead to an algorithm for plotting the "interesting parts" of the curve, i.e. the parts showing the main geometrical features of it. On the theoretical side, we prove that these curves, with the exceptions of lines and circles, cannot be algebraic (in cartesian coordinates), we characterize the existence of infinitely many self-intersections, and we connect this with certain phenomena which are not possible in the algebraic world, namely the existence of limit circles, limit points, or spiral branches. On the practical side, we provide an algorithm which has been implemented in the computer algebra system Maple to visualize this kind of curves. Our implementation makes use (and improves some aspects of) the command polarplot currently available in Maple for plotting curves in polar form.