Primitivoids and inversions of plane curves (Q2181694)

The authors continue their investigation of curves in the Euclidean plane \(\mathbb R^2\) that arise from a given curve. In a previous paper [the authors, Note Mat. 39, No. 2, 13--24 (2019; Zbl 1433.53005)], they dealt with the relation between pedaloids and evolutoids, which are generalisations of the pedal curve (with respect to the origin) and the evolute, respectively. At each point of the given curve (which is assumed not to pass through the origin), there is a unique line that is normal to the point's position vector. The envelope of all such lines is the primitive of the original curve. So, forming the pedal and the primitive are inverse operations. After introducing the notion of anti-pedal, it is shown that the primitive is equal to the anti-pedal of that curve which arises from the original curve under the inversion at the unit circle centred at the origin. As for the pedal, the singularities of the antipedal correspond to the inflection points of the original curve. Furthermore, various types of primitivoids of a plane curve are introduced; these are siblings of the primitive. It turns out that the shapes of all primitivoids of a given curve are similar. The paper is rounded off with a number of further results and closes with several impressive illustrations.