A complete characterization of Pythagorean-hodograph preserving mappings

Abstract – We fully characterize the mappings (Formula presented) 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 (Formula presented), where (Formula presented) is meromorphic and satisfies (Formula presented) at every pole. In higher dimensions, PH-preservation forces (Formula presented) to be a conformal map; for (Formula presented), 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 (Formula presented) 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.