Low Degree Euclidean and Minkowski Pythagorean Hodograph Curves

Abstract

In our contribution we study cubic and quintic Pythagorean Hodograph (PH) curves in the Euclidean and Minkowski planes. We analyze their control polygons and give necessary and sufficient conditions for cubic and quintic curves to be PH. In the case of Euclidean cubics the conditions are known and we provide a new proof. For the case of Minkowski cubics we formulate and prove a new simple geometrical condition. We also give conditions for the control polygons of quintics in both types of planes. Moreover, we introduce the new notion of the preimage of a transformation, which is closely connected to the so-called preimage of a PH curve. We determine which transformations of the preimage curves produce similarities of PH curves in both Euclidean and Minkowski plane. Using these preimages of transformations we provide simple proofs of the known facts that up to similarities there exists only one Euclidean PH cubic (the so-called Tschirnhausen cubic) and two Minkowski PH cubics. Eventually, with the help of this novel approach we classify and describe the systems of Euclidean and Minkowski PH quintics.