Minimal Surfaces via Complex Quaternions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915773367189504 |
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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 | Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstraß-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal Surfaces via Complex Quaternions Altavilla, Amedeo Schröcker, Hans-Peter Šír, Zbyněk Vršek, Jan Complex Variables Rings and Algebras 53A10 16H05 46S05 65D17 Minimal surfaces play a fundamental role in differential geometry, with applications spanning physics, material science, and geometric design. In this paper, we explore a novel quaternionic representation of minimal surfaces, drawing an analogy with the well-established theory of Pythagorean Hodograph (PH) curves. By exploiting the algebraic structure of complex quaternions, we introduce a new approach to generating minimal surfaces via quaternionic transformations. This method extends classical Weierstraß-Enneper representations and provides insights into the interplay between quaternionic analysis, PH curves, and minimal surface geometry. Additionally, we discuss the role of the Sylvester equation in this framework and demonstrate practical examples, including the construction of Enneper surface patches. The findings open new avenues in computational geometry and geometric modeling, bridging abstract algebraic structures with practical applications in CAD and computer graphics. |
| title | Minimal Surfaces via Complex Quaternions |
| topic | Complex Variables Rings and Algebras 53A10 16H05 46S05 65D17 |
| url | https://arxiv.org/abs/2504.17377 |