Untangling Surfaces via Shape and Mesh Repulsion

Self-intersections are widespread in surface meshes and invalidate downstream simulation, fabrication, and learning pipelines. Existing approaches typically treat self-intersections as local collision events, but embeddedness (i.e., lack of self-intersections) is a global geometric property that cannot be enforced through local reasoning alone. We introduce an energy-based framework that enforces surface embeddedness simultaneously at the shape and mesh levels, based on the insight that successful untangling requires accounting for both global shape-level interactions and local mesh-level interactions. A shape-level energy captures global entanglement independent of discretization, while a mesh-level penalty regularizes local discrete interactions. Together, these energies enable reliable removal of self-intersections without changing mesh connectivity and apply to a broad class of geometries, including surfaces with boundary, non-manifold configurations, immersion failures, and multi-object scenes. Compared to prior state-of-the-art methods, our approach resolves self-intersections across challenging datasets, enabling reliable downstream processing of surface meshes.