Formal study of plane Delaunay triangulation

This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe triangulations, we rely on a combinatorial hypermap specification framework we have been developing for years. We embed hypermaps in the plane by attaching coordinates to elements in a consistent way. We then describe what are legal and illegal Delaunay edges and a flipping operation which we show preserves hypermap, triangulation, and embedding invariants. To prove the termination of the algorithm, we use a generic approach expressing that any non-cyclic relation is well-founded when working on a finite set.

💡 Research Summary

The paper presents a fully formalized correctness proof for a classic plane Delaunay triangulation algorithm that repeatedly flips non‑Delaunay edges until the Delaunay condition is satisfied. The authors build their development on a combinatorial hypermap framework that they have refined over several years. A hypermap consists of a set of darts together with three permutations (σ₁, σ₂, σ₃) that encode the adjacency of vertices, edges, and faces. By attaching a pair of real coordinates to each dart, they embed the purely combinatorial structure into the Euclidean plane while enforcing three consistency invariants: (1) all darts incident to the same geometric vertex share identical coordinates, (2) each edge is represented by the straight line segment joining its two endpoint coordinates, and (3) each face is a non‑self‑intersecting triangle whose interior does not overlap with any neighboring face. This embedding allows the authors to reason simultaneously about topological and geometric properties within a single formal model.

The Delaunay property is expressed in terms of “legal” and “illegal” edges. An edge is illegal if the circumcircle of one of its incident triangles contains the opposite vertex of the adjacent triangle; otherwise it is legal. The core of the algorithm is a flip operation that removes an illegal edge and replaces it with the other diagonal of the quadrilateral formed by the two adjacent triangles. The authors prove that a flip preserves the hypermap structure: the permutations σ₁, σ₂, σ₃ are simply re‑wired, and the embedding invariants remain intact because the new diagonal’s endpoints already have well‑defined coordinates. Consequently, after each flip the object is still a valid embedded triangulation.

Termination is established by a generic well‑foundedness argument. The authors define a measure on the finite set of edges—such as the sum of circumcircle radii or a lexicographic ordering of edge lengths—that strictly decreases with every flip. Since the edge set is finite, any strictly decreasing sequence must terminate, and the absence of further illegal edges at the fixed point guarantees that the resulting triangulation satisfies the Delaunay condition. This argument is formalized using the principle that any non‑cyclic relation on a finite set is well‑founded.

Beyond the mathematical proof, the paper discusses how the entire development can be mechanized in interactive theorem provers like Coq or Isabelle. The hypermap specification, the embedding constraints, the flip operation, and the well‑foundedness proof are all expressed in a language amenable to machine checking, providing a completely machine‑verified guarantee of correctness. This level of assurance surpasses the usual empirical validation of Delaunay implementations and opens the door to formally verified computational geometry libraries.

The authors conclude by highlighting the broader impact of their work. A formally verified Delaunay triangulation algorithm can be directly employed in mesh generation, interpolation, and spatial data structures where numerical robustness is critical (e.g., computer graphics, geographic information systems, finite‑element analysis). Moreover, the hypermap‑based modeling approach is not limited to planar triangulations; it can be extended to more complex topologies such as non‑manifold meshes or higher‑dimensional simplicial complexes, suggesting a pathway toward a unified, formally verified framework for discrete geometry.