On a theorem of O. Röschel in planar isotropic curve theory (Q1818911)

The author uses curves of constant curvature \(\kappa_G\) with respect to the equiform group of the isotropic plane. All curves of this class have an exceptional point (pole, inflection point \(\dots\)). He studies all curves of given constant curvature \(\kappa_G (\neq 1, 3/2)\) osculating a given curve \(c\) in a general point \(P\). The author shows that the exceptional points of all these osculating curves are situated on an isotropic circle \(k(P, \kappa_G)\). This result generalizes the isotropic analogon to the circle of ABRAMESCU. A further part of the paper deals with curves of constant equiform curvature \(\kappa_G\) hyperosculating \(c\) at \(P\). For variable \(\kappa_G\) the exceptional points of these curves are situated on a special isotropic hyperbola. Results on the isotropic counterparts to Stammler's circles connected with a triangle finish the paper.