Solving Euclidean Problems by Isotropic Initialization
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908389636833280 |
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| author | Yorov, Khusrav Wang, Bolun Skopenkov, Mikhail Pottmann, Helmut Jiang, Caigui |
| author_facet | Yorov, Khusrav Wang, Bolun Skopenkov, Mikhail Pottmann, Helmut Jiang, Caigui |
| contents | Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous problems in isotropic geometry. Isotropic geometry can be viewed as a structure-preserving simplification of Euclidean geometry. The solutions found in the isotropic case give insight and can initialize optimization algorithms to solve the original Euclidean problems. We illustrate this general approach with three examples: quad-mesh mechanisms, composite asymptotic-geodesic gridshells, and asymptotic gridshells with constant node angle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solving Euclidean Problems by Isotropic Initialization Yorov, Khusrav Wang, Bolun Skopenkov, Mikhail Pottmann, Helmut Jiang, Caigui Computational Geometry Differential Geometry Metric Geometry 53A10, 53A40, 53A70 G.2 Many problems in Euclidean geometry, arising in computational design and fabrication, amount to a system of constraints, which is challenging to solve. We suggest a new general approach to the solution, which is to start with analogous problems in isotropic geometry. Isotropic geometry can be viewed as a structure-preserving simplification of Euclidean geometry. The solutions found in the isotropic case give insight and can initialize optimization algorithms to solve the original Euclidean problems. We illustrate this general approach with three examples: quad-mesh mechanisms, composite asymptotic-geodesic gridshells, and asymptotic gridshells with constant node angle. |
| title | Solving Euclidean Problems by Isotropic Initialization |
| topic | Computational Geometry Differential Geometry Metric Geometry 53A10, 53A40, 53A70 G.2 |
| url | https://arxiv.org/abs/2506.01726 |