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-)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.
From: Amedeo Altavilla [view email]
[v1]
Mon, 22 Dec 2025 17:14:03 UTC (599 KB)
[v2]
Tue, 23 Dec 2025 11:15:22 UTC (599 KB)
[v3]
Thu, 11 Jun 2026 18:43:55 UTC (598 KB)
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