Minimum feature size preserving decompositions

TL;DR

This paper introduces an algorithm for quadrangulating polygons with minimal feature size degradation, using Steiner points, and establishes lower bounds showing constant degradation isn't always achievable.

Contribution

It presents a novel quadrangulation algorithm with constant degradation and proves lower bounds on the limitations of triangulation regarding feature size preservation.

Findings

Quadrangulation can be achieved with Theta(n) Steiner points and constant degradation.

Lower bounds show some polygons cannot be triangulated with constant degradation, even with unlimited Steiner points.

The method ensures faces are quadrangular, but the number of edges per face may vary.

Abstract

The minimum feature size of a crossing-free straight line drawing is the minimum distance between a vertex and a non-incident edge. This quantity measures the resolution needed to display a figure or the tool size needed to mill the figure. The spread is the ratio of the diameter to the minimum feature size. While many algorithms (particularly in meshing) depend on the spread of the input, none explicitly consider finding a mesh whose spread is similar to the input. When a polygon is partitioned into smaller regions, such as triangles or quadrangles, the degradation is the ratio of original to final spread (the final spread is always greater). Here we present an algorithm to quadrangulate a simple n-gon, while achieving constant degradation. Note that although all faces have a quadrangular shape, the number of edges bounding each face may be larger. This method uses Theta(n) Steiner points and produces Theta(n) quadrangles. In fact to obtain constant degradation, Omega(n) Steiner points are required by any algorithm. We also show that, for some polygons, a constant factor cannot be achieved by any triangulation, even with an unbounded number of Steiner points. The specific lower bounds depend on whether Steiner vertices are used or not.