Classification of planar Pythagorean hodograph curves

Abstract

We classify planar polynomial Pythagorean-hodograph curves of any degree with respect to Euclidean similarities. We also analyze possible global shapes of the Pythagorean-hodograph curves of degree four and five and describe all possible configurations of their singular points. This description results in a classification of Pythagorean-hodograph quartics and quintics with respect to the homeomorphisms of the plane. We also describe all the Pythagorean-hodograph curves of any degree up to five which exhibit central or axial symmetry.