Meshes Preserving Minimum Feature Size

The minimum feature size of a planar straight-line graph is the minimum distance between a vertex and a nonincident edge. When a polygon is partitioned into a mesh, the degradation is the ratio of original to final minimum feature size. We show that some planar straight-line graphs cannot be triangulated with constant degradation, even with an unbounded number of Steiner points and triangles. This result answers a 14-year-old open problem by Bern, Dobkin, and Eppstein. For an n-vertex input, we obtain matching worst-case lower and upper bounds on degradation of Θ(lgn). Our upper bound comes from a new meshing algorithm that uses O(n) triangles and O(n) Steiner points. If we allow triangles to have Steiner points along their sides, a construction is presented that achieves O(1) degradation.