Computing pseudotriangulations via branched coverings

We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility complexes and on the extension of that theory to the setting of branched coverings. The problem of computing a pseudotriangulation that contains a given set of bitangent line segments is also examined.

💡 Research Summary

The paper tackles the problem of constructing a pseudotriangulation for a finite planar family of pairwise‑disjoint convex bodies when the only input available is the family’s chirotope, i.e., a combinatorial encoding of the relative positions of the bodies. A chirotope for convex bodies extends the classical notion for point sets: for each ordered triple of indices it records twenty Boolean predicates describing on which side of each of the four possible directed bitangents (left‑left, left‑right, right‑left, right‑right) the third body lies, and whether it meets the initial, median, or final part of the bitangent. This purely order‑type information suffices to reconstruct the entire visibility structure of the family without any metric data.

The authors develop a three‑stage algorithm that runs in O(n log n) time and uses linear space. The first stage computes the convex hull of the bodies, i.e., the set of boundary bitangent segments, by adapting classic convex‑hull techniques to the chirotope setting. The second stage builds a cross‑section of the visibility complex of the free space (the complement of the interiors of the bodies, possibly cut by a set of constraints). The visibility complex is a cell complex whose 0‑cells are free bitangents, 1‑cells are maximal continuous families of lines that avoid the bodies, and 2‑cells are regions bounded by two bitangents. By passing to a branched covering of this complex, each bounded 2‑cell acquires a unique source and sink vertex, allowing the definition of a partial order on cells. Selecting a maximal antichain in this order yields a directed multigraph (the cross‑section) whose arcs correspond to 2‑cells and whose vertices are the 0‑ and 1‑cells.

The third stage applies a greedy procedure to the cross‑section: bitangent segments are examined in the order induced by the antichain, and a segment is added to the output pseudotriangulation if and only if it separates the current free space into two components. Because the decision depends only on the combinatorial structure of the visibility complex, each test is O(1). The greedy process produces a maximal set of interior non‑crossing bitangents, i.e., a pseudotriangulation, and automatically includes the boundary bitangents found in the first stage.

The algorithm also handles constrained pseudotriangulations, where a prescribed set of interior bitangents must appear in the final structure. The authors prove that if the number of constrained bitangents that appear consecutively on the boundary of any pseudotriangle is bounded by a constant (a condition satisfied in many practical scenarios), the same O(n log n) time bound holds.

Technical contributions include a detailed treatment of visibility complexes on topological planes, the introduction of branched coverings to regularize the complex, and a perturbation scheme that converts non‑simple chirotopes (those containing tritangents) into simple ones without changing the set of admissible bitangents. The paper’s appendices provide (i) a duality theory for topological planes, (ii) algorithms for computing visibility graphs of line‑segment sets, (iii) a reduction of the point‑set pseudotriangulation problem to the convex‑body setting, and (iv) discussions of extensions to non‑Euclidean planes.

In summary, the work presents a novel, purely combinatorial framework for pseudotriangulation of convex bodies, achieving optimal asymptotic performance while relying only on order‑type information. It opens avenues for further research on dynamic updates, higher‑dimensional generalizations, and applications in motion planning, geometric optimization, and computational topology.