EJMT Abstract

Geometric loci are an important topic in geometry. We explore constructs of plane curves and of space curves inspired by space trajectories (bicircular and tricircular motions). The exploration relies on joint usage of Dynamic Geometry Software (DGS) and of a Computer Algebra System (CAS), and involves automated methods to determine loci and sometimes dichotomy methods. The computer assisted work reveals some surprises, in particular the respective roles of the two kinds of software may be different from what previous works revealed. If previously the roles were quite distinct, DGS for exploration and numerical approach, and CAS for symbolic proof, here the CAS is needed for numerical methods also. The curves are described by trigonometric parametrization, and implicitization is performed using elimination, for 2D and 3D models. Finally, we discuss approaches to developments using models and elaborate on the importance to develop automatic dialog between kinds of software.