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.

翻译：曲面点离散化广泛应用于从物体渲染到曲面偏微分方程求解等众多领域。这些应用通常要求采样点兼具局部正则性与全局曲率自适应性，以维持算法鲁棒性与计算效率。即使对于简单解析曲面，计算数值良态的离散点布局也并非易事。本文提出一种算法，可在曲面上根据预设长度场寻找近最优的点分布。该算法通过近似最小化点间局部相互作用的全局势能实现优化，采用梯度下降法结合线搜索确定最优步长进行求解。我们使用水平集方法描述曲面，无需额外曲面吸引力即可完成所有必要投影操作。为加速收敛，算法通过积分支持度量动态检测局部点过剩或缺失，并执行点融合与插入操作。我们在参数曲面与非参数曲面等多种几何形状上测试该算法，计算了具有不同曲率自适应性的点分布。实验表明，该算法局部平均偏差可精准控制在预设目标间距范围内。总体而言，所提算法能够快速稳健地收敛至最终的曲面点数量与分布。