A complete characterization of Pythagorean-hodograph preserving mappings

Abstract

We fully characterize the mappings Φ that send every Pythagorean-hodograph (PH) curve to a PH curve. We prove that in any dimension, such mappings are precisely the conformal functions whose dilation is the square of a real rational function. In the planar case, this implies (up to conjugation) that ∂Φ/∂z=Ψ2, where Ψ is meromorphic and satisfies Res⁡(Ψ2)=0 at every pole. In higher dimensions, PH-preservation forces Φ to be a conformal map; for n≥3, Liouville’s theorem then implies that any local diffeomorphism with this property is (anti-)Möbius. These results subsume the previously known ‘scaled PH-preserving’ constructions of mappings R2→R3 and align with Ueda’s conformal viewpoint on isothermal and spherical geometries. At the level of examples, we demonstrate how PH-preserving mappings relate to the construction of rational PH curves and minimal surfaces.