Happy endings for flip graphs

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio

Subquadratic nonobtuse triangulation of convex polygons

A convex polygon with n sides can be triangulated by O(n^1.85) triangles, without any obtuse angles. The construction uses a novel form of geometric divide and conquer

Cubic Partial Cubes from Simplicial Arrangements

We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure

Linear Complexity Hexahedral Mesh Generation

We show that any polyhedron forming a topological ball with an even number of quadrilateral sides can be partitioned into O(n) topological cubes, meeting face to face. The result generalizes to non-simply-connected polyhedra satisfying an additional bipartiteness condition. The same techniques can also be used to reduce the geometric version of the hexahedral mesh generation problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at the 12th ACM Symp. on Computational Geometry. This is the final version, and will appear in a special issue of Computational Geometry: Theory and Applications for papers from SCG '9