Abstract
We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings.
Highlights
The straight skeleton is a skeletal structure of a polygon P, similar to the Voronoi diagram
By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist
We can answer the question to the affirmative, that is, we show how to safely define weighted straight skeletons in the presence of multiple simultaneous, co-located split events. (Note that due to the discontinuous character of straight skeletons, it is not possible to tackle this problem by means of simulation of simplicity.4) We first rephrase this problem as a planar matching problem of directed pseudo-lines and show how to transform the planar matching problem into a stable roommates problem
Summary
The straight skeleton is a skeletal structure of a polygon P , similar to the Voronoi diagram. Biedl et al.[10] showed that basic properties of unweighted straight skeletons do not carry over to weighted straight skeletons in general They proposed solutions for an ambiguity in the definition of straight skeletons caused by certain edge events. Planar Matchings for Weighted Straight Skeletons 213 one fundamental principle: Between events, the wavefront is a planar collection of wavefront polygons This is achieved when handling edge events and “simple” split events where at most four edges are in the wavefront afterwards. We prove that our particular stable roommates problem always possesses a solution Such a solution tells us how to do the event handling of the wavefront in order to maintain planarity
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