A Complete Characterization of Pythagorean Hodograph Preserving Mappings

TL;DR

This paper fully characterizes the class of conformal mappings that preserve Pythagorean-hodograph (PH) curves across all dimensions, revealing they are essentially (anti-)M"obius transformations in higher dimensions.

Contribution

It provides a complete mathematical description of PH-preserving mappings, extending known planar results to higher dimensions and connecting them with conformal and M"obius transformations.

Findings

PH-preserving mappings are conformal functions with squared rational dilation.

In 2D, such mappings relate to meromorphic functions with zero residues at poles.

In dimensions three and higher, PH-preserving maps are (anti-)M"obius transformations.

Abstract

We fully characterize the mappings $\Phi$ 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 $\partial\Phi/\partial z = \Psi^{2}$, where $\Psi$ is meromorphic and satisfies $\operatorname{Res}(\Psi^{2}) = 0$ at every pole. In higher dimensions, PH preservation forces $\Phi$ to be a conformal map; for $n \ge 3$, Liouville's theorem then implies that any local diffeomorphism with this property is (anti-)M\"obius. 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.