Solving surface equations by improving geometric projections and embeddings
Walk-on-Ambient-Patch: Solver-Adapted Isometric Embeddings for Projected Walk on Spheres
Graphics
Summary
Solving equations on curved surfaces can be tricky when surfaces fold or overlap in complex ways, making it hard to move smoothly along them. The authors propose a new method that changes how the surface is represented and sampled, helping avoid confusing overlaps and making calculations more accurate. Their approach keeps the surface's true shape while allowing some extra mathematical dimensions to separate tricky parts. Tests show this method lessens problems with projection conflicts and maintains accurate shapes better, helping solve surface problems more reliably.
What this means in practice
- •For computational physics teams: Solve partial differential equations on complex folded surfaces with improved accuracy from adaptive surface embedding.
- •For computer graphics developers: Generate more reliable surface simulations by reducing projection conflicts in curved and folded geometries during rendering.
Authors
Yunfeng Liao, Xiucheng Li
Abstract
Projected Walk on Spheres (PWoS) solves surface PDEs through ambient sampling and closest-point projection, but tightly folded embeddings can force short steps or ambiguous projections despite simple intrinsic geometry. We introduce Walk-on-Ambient-Patch (WoAP), which treats the embedding as a solver variable. Edge-isometry constraints and tangent-point repulsion separate intrinsically distant regions, optionally in higher codimension; patch states track residual projection branches without introducing PDE boundaries. We analyze the resulting geometry--sampling tradeoff: additional dimensions provide separation freedom but reduce uniform tangent progress, while strong tangent-biased proposals incur large importance-weight moments. Folded-surface experiments demonstrate reduced projection conflicts, a separate hard-constraint pilot verifies metric preservation, and an internal fixed-RMSE comparison favors higher dimensions at stricter accuracy targets. Together, these results motivate choosing the embedding and projection state jointly with the stochastic solver.