Globally adaptive and locally regular point discretization of curved surfaces

2026-05-05

AI Summary

This work addresses the challenge of achieving both global curvature adaptivity and local regularity in surface point discretization by proposing an optimization method based on global energy minimization. The approach represents surfaces implicitly via level sets and detects local point density deviations using integral-based support measures. By combining gradient descent with line search, it dynamically adjusts point distributions and enables efficient projection without requiring additional surface attraction forces. The algorithm supports dynamic point merging and insertion to accelerate convergence and is applicable to both parametric and non-parametric surfaces. Experiments demonstrate that the method rapidly generates highly robust and accurate point distributions on a variety of complex surfaces, exhibiting minimal local spacing deviation and effectively balancing curvature adaptivity with regularity, thereby meeting the numerical stability and efficiency requirements of applications such as rendering and surface-based partial differential equation solvers.

Abstract

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.

Problem

• point discretization
• curved surfaces
• local regularity
• curvature adaptivity
• surface sampling

Innovation

• curvature-adaptive discretization
• point distribution optimization
• level-set surface representation
• dynamic point insertion/fusion
• gradient descent with line search