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.