An automated study of isoptic curves of an astroid

Abstract

Learning mathematics in a technology rich environment enables to revive classical topics which have been removed from the curriculum a long time ago. Theoretical issues and their applications can be studied within an experimental process, using automated proofs. We present a technology based study of isoptic curves of an astroid, which is a non smooth and non convex curve. We take advantage of both algebraic symbolic features, such as resultants and Gröbner bases packages implemented in a Computer Algebra System (CAS), together with a Dynamical Geometry System (DGS). The usage of algebraic methods is made possible by the translation of the trigonometric parametrization into rational expressions, and then into polynomial equations. The dynamics provided by the DGS enable the discovery of special features of the isoptics, in particular the fact that the isoptic has singular points and, according to the viewing angle, the isoptic may be inscribed in the astroid or not.