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 $\partialΦ/\partial z = Ψ^{2}$, where $Ψ$ is meromorphic and satisfies $\operatorname{Res}(Ψ^{2}) = 0$ at every pole. In higher dimensions, PH preservation forces $Φ$ to be a conformal map; for $n \ge 3$, 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 $\mathbb{R}^2 \to \mathbb{R}^3$ 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.