A Complete Characterization of Pythagorean Hodograph Preserving Mappings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915692195872768 |
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| author | Altavilla, Amedeo Schröcker, Hans-Peter Šír, Zbyněk Vršek, Jan |
| author_facet | Altavilla, Amedeo Schröcker, Hans-Peter Šír, Zbyněk Vršek, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Complete Characterization of Pythagorean Hodograph Preserving Mappings Altavilla, Amedeo Schröcker, Hans-Peter Šír, Zbyněk Vršek, Jan Differential Geometry Complex Variables Metric Geometry Primary 65D17, 53A30, Secondary 30C35 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. |
| title | A Complete Characterization of Pythagorean Hodograph Preserving Mappings |
| topic | Differential Geometry Complex Variables Metric Geometry Primary 65D17, 53A30, Secondary 30C35 |
| url | https://arxiv.org/abs/2512.19587 |