Pythagorean hodograph curves of degree four

Abstract

All Euclidean similitudes and reparameterizations of planar curves with Pyhtagorean Hodograph (PH curves) can be induced by Euclidean similitudes and reparameterizations of their preimage. We exploit this fact in order to describe the explicit and implicit form of all planar PH curves of degree 4.

We also classify these curves with respect to the type and distribution of their singularities. Our results are illustrated by examples.

Keywords

• Euclidean similitudes
• curves with Pythagoream Hodograph
• curves of degree 4
• algebraic curves
• singularity
• genus