Singular points of offsets of plane curves: man-and-machine collaboration with CAS, DGS and AI

We explore envelopes of families of circles of constant radius centered on a plane curve, and offsets of plane curves, using automated commands for envelopes and geometric loci. The topology of envelopes and offsets is generally more complicated than that of the progenitor, so it is important to study the singularities.These can be cusps and crunodes (self-intersections). The work involves both analytic and algebraic methods (such as elimination), networking between different kinds of software, and automated commands to determine envelopes and geometric loci. Accurate plotting is a central issue, because in a neighborhood of a singular point the software may have difficulties and leave a blank gap. To address this issue, we demonstrate how the new Plot2D command provided byGeoGebra Discovery is effective. Finally, we analyze the output of generative AI for the above questions and discuss briefly how the pitfalls can and should be used to develop critical thinking and other skills. Generative AI can nevertheless be useful for rapid paraphrasing, suggesting candidate parametrizations, and proposing computational steps, provided that all results are verified by theorem based software and mathematical reasoning. The activities presented here have been proposed to undergraduates, both in regular classes and as individual students acting as undergraduate researchers, and to in-service teachers working toward an advanced degree.