Envelopes and Offsets of Two Algebraic Plane Curves: Exploration of Their Similarities and Differences

Several non equivalent definitions exist for the envelope of a 1-parameter family of plane curves. Another notion, often considered as related to envelopes, is the offset at a given distance of a plane curve. Using the so-called analytic definition, we study and compare the envelope of a 1-parameter family of circles centered on a parabola and an offset of this parabola. Then we perform a similar study for an astroid. This illustrates the non equivalence of the two notions. The work is performed with networking with a Computer Algebra System and a Dynamical Geometry System, bringing them into a certain form of dialog. The obtained curves are sextics, and their discovery as envelopes leads to an automated exploration of Talbot curves. Another output shows these curves within a unifying framework involving a Maltese Cross.