Topological Offsets

TL;DR

This work introduces topological offsets, a topology-preserving approach to generate manifold, watertight, self-intersection-free offset surfaces that strictly enclose the input. By embedding the input in a background tetrahedral mesh and using a purely topological construction, the method guarantees a surface with the same topology as an infinitesimal offset, while enabling optional conversion to finite offsets and layered configurations. The pipeline comprises input preprocessing, a simplicial embedding, offset insertion, and an optional optimization phase that preserves topology via rigorous invariants and exact predicates. The results on Thingi10k demonstrate robustness across non-manifold inputs, self-intersections, and complex topologies, with applications including non-manifold repair, layered offsets, and reliable finite-offset computation, accompanied by an open-source reference implementation. This work provides a principled, topology-first offset framework that broadens offset applications in graphics and CAD while guaranteeing enclosures and intersection-free outputs.

Abstract

We introduce Topological Offsets, a novel approach to generate manifold and self-intersection-free offset surfaces that are topologically equivalent to an offset infinitesimally close to the surface. Our approach, by construction, creates a manifold, watertight, and self-intersection-free offset surface strictly enclosing the input, while doing a best effort to move it to a prescribed distance from the input. Differently from existing approaches, we embed the input in a background mesh and insert a topological offset around the input with purely combinatorial operations. The topological offset is then inflated/deflated to match the user-prescribed distance while enforcing that no intersections or non-manifold configurations are introduced. We evaluate the effectiveness and robustness of our approach on the Thingi10k dataset, and show that topological offsets are beneficial in multiple graphics applications, including (1) converting non-manifold surfaces to manifold ones, (2) creating layered offsets, and (3) reliably computing finite offsets.

Figures (40)

• Figure 1: Our method generates manifold meshes with topological guarantees even if the input (left) is open, non-manifold, non-orientable, and self-intersecting. The right image is a cut-view of the offset.
• Figure 2: Topological offsets (blue) always have the same topology for any given input (white), while the topology of finite offsets (green) depend on the offset distance
• Figure 3: The corner between the two boxes does not have a unique vertex normal, i.e., there is no direction in which the vertex can be offset without intersecting the input. Our method does not rely on vertex normals and can, therefore, handle this case properly. See Jiang2020 for a more detailed discussion.
• Figure 4: Our method utilizes local subdivisions instead of a uniform refinement, leading to merely a fraction of triangles. Both, our method and guo2024robust, are guaranteed to construct an offset that is homeomorphic to a closed and manifold input mesh without self-intersections.
• Figure 5: An infinitesimal offset family (blue curves). If $\epsilon$ becomes too large, the offset changes topology (red curve). We are interested in a topological offset (green curve) that admits a continuous bijective map to the infinitesimal one.

Theorems & Definitions (22)

• Definition 1: $\hat{\epsilon}$ Infinitesimal Offset Family
• Theorem 1: Infinitesimal Offsets
• proof
• Definition 2: Simplicial Embedding
• Lemma 1: Offset Locality
• proof
• Lemma 2
• proof
• Lemma 3
• proof