# Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions

@article{Eppstein1999RaisingRC, title={Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions}, author={David Eppstein and Jeff Erickson}, journal={Discrete \& Computational Geometry}, year={1999}, volume={22}, pages={569-592} }

Abstract. The straight skeleton of a polygon is a variant of the medial axis introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an n -gon with r reflex vertices in time O(n1+ε + n8/11+εr9/11+ε) , for any fixed ε >0 , improving the previous best upper bound of O(nr log n) . Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process…

## 60 Citations

Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions

- Computer Science, MathematicsSCG '98
- 1998

The straight skeleton of an n -gon with r reflex vertices is constructed in time O(n 1+e + n 8/11+e r 9/11-e ) , for any fixed e >0, improving the previous best upper bound of O(nr log n) .

Computing Mitered Offset Curves Based on Straight Skeletons

- Computer Science
- 2014

This work extends and adapt Aichholzer and Aurenhammer's triangulation-based straight-skeleton algorithm to make it process real-world data on a conventional finite-precision arithmetic and demonstrates the practical suitability of using straight skeletons for the offsetting of complex PSLGs.

On the Structure of Straight Skeletons

- Mathematics, Computer ScienceICCSA Workshops
- 2008

It is shown that each Mi is a pruned medial axis for a certain convex polygon Qi closely related to G, and an optimal algorithm for computation of all those polygons is given.

On the Structure of Straight Skeletons

- Mathematics, Computer Science2008 International Conference on Computational Sciences and Its Applications
- 2008

It is shown that each M-sub i is a pruned medial axis for a certain convex polygon Q<sub>i</sub> closely related to G, and an optimal algorithm for computation of all those polygons is given.

Generating Realistic Roofs over a Rectilinear Polygon

- Mathematics, Computer ScienceISAAC
- 2011

An algorithm is presented that enumerates a combinatorial representation of each such roof in O(1) time per roof without repetition, after O(n4) preprocessing time.

Linear transformation distance for bichromatic matchings

- Mathematics, Computer ScienceComput. Geom.
- 2018

An alternative proof for the connectivity of the transformation graph of BR-matchings is provided and an upper bound of 2 n for its diameter is proved, which is asymptotically tight.

Computing Motorcycle Graphs Based on Kinetic Triangulations

- Mathematics, Computer ScienceCCCG
- 2012

An ecient algorithm for computing generalized motorcycle graphs, in which motorcycles are allowed to emerge after time zero, which constitutes a signicant practical improvement over the motorcycle code Moca, which runs in O(n logn) time only if the motorcycles are distributed uniformly enough.

Min-/Max-Volume Roofs Induced by Bisector Graphs of Polygonal Footprints of Buildings

- Computer Science, MathematicsInt. J. Comput. Geom. Appl.
- 2018

This paper shows how to construct a roof over the polygonal footprint of a building that has minimum or maximum volume among all roofs that drain water and extends the standard plane-sweep approach known from the theory of straight skeletons by additional events.

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