Globally adaptive and locally regular point discretization of curved surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911648898351104 |
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| author | Schulze, Lennart J. Sbalzarini, Ivo F. |
| author_facet | Schulze, Lennart J. Sbalzarini, Ivo F. |
| contents | Point discretization of curved surfaces is required in many applications ranging from object rendering to the solution of surface partial differential equations (PDEs). These applications often impose that surfaces are sampled with local regularity and global curvature adaptivity to maintain robustness and efficiency. Computing numerically well-conditioned point discretization is non-trivial, even for simple analytic curved surfaces. We present an algorithm for finding near-optimal surface point distributions governed by a prescribed length field on curved surfaces. The algorithm works by approximately minimizing a global potential over local point-point interactions. The optimization problem is solved using gradient descent, accelerated by line search to find optimal step sizes. We use a level-set method to describe the surface and perform all required projections without requiring additional surface-attractive forces. To further accelerate convergence, the algorithm dynamically fuses and inserts points where a local excess or lack of points is detected using an integral support measure. We test the proposed algorithm on a variety of shapes, ranging from parametric to non-parametric surfaces. We compute point distributions with different curvature adaptivity and show that the algorithm achieves low average deviation from the prescribed target spacing locally. Overall, the presented algorithm rapidly and robustly converges to the final number and distribution of surface points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03803 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Globally adaptive and locally regular point discretization of curved surfaces Schulze, Lennart J. Sbalzarini, Ivo F. Computational Engineering, Finance, and Science Graphics Point discretization of curved surfaces is required in many applications ranging from object rendering to the solution of surface partial differential equations (PDEs). These applications often impose that surfaces are sampled with local regularity and global curvature adaptivity to maintain robustness and efficiency. Computing numerically well-conditioned point discretization is non-trivial, even for simple analytic curved surfaces. We present an algorithm for finding near-optimal surface point distributions governed by a prescribed length field on curved surfaces. The algorithm works by approximately minimizing a global potential over local point-point interactions. The optimization problem is solved using gradient descent, accelerated by line search to find optimal step sizes. We use a level-set method to describe the surface and perform all required projections without requiring additional surface-attractive forces. To further accelerate convergence, the algorithm dynamically fuses and inserts points where a local excess or lack of points is detected using an integral support measure. We test the proposed algorithm on a variety of shapes, ranging from parametric to non-parametric surfaces. We compute point distributions with different curvature adaptivity and show that the algorithm achieves low average deviation from the prescribed target spacing locally. Overall, the presented algorithm rapidly and robustly converges to the final number and distribution of surface points. |
| title | Globally adaptive and locally regular point discretization of curved surfaces |
| topic | Computational Engineering, Finance, and Science Graphics |
| url | https://arxiv.org/abs/2605.03803 |