Pythagorean-hodograph curves of Tschirnhaus type are sinusoidal spirals

Recently, Bizzarri et al. (2021) discussed the so-called Pythagorean-hodograph curves of Tschirnhaus type, a generalization to higher degrees of Tschirnhausen cubic. We recall that these curves in Bezier form coincide with the typical curves introduced by Mineur et al. (1998), as well as with a classical family of sinusoidal spirals. Therefore, they all enjoy the same properties, such as the rational character of their offsets or the existence of only one curve (up to similarities) for each degree. By elucidating this connection among curves of Tschirnhaus type, typical curves, and sinusoidal spirals, we rederive several relevant results found by Bizzarri et al. (2021). (C) 2022 The Author(s). Published by Elsevier B.V.