Representing Directed Trees as Straight Skeletons
The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process.
In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.
KeywordsVoronoi Diagram Simple Polygon Interior Node Geometric Graph Interior Angle
- 1.Aichholzer, O., Aurenhammer, F.: Straight skeletons for general polygonal figures in the plane. In: Samoilenko, A. (ed.) Voronoi’s Impact on Modern Sciences II, vol. 21, pp. 7–21. Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, Ukraine (1998)Google Scholar
- 2.Aichholzer, O., Aurenhammer, F., Alberts, D., Gärtner, B.: A novel type of skeleton for polygons. J. Univ. Comput. Sci. 1(12), 752–761 (1995)Google Scholar
- 3.Aichholzer, O., Biedl, T., Hackl, T., Held, M., Huber, S., Palfrader, P., Vogtenhuber, B.: Representing directed trees as straight skeletons [cs.CG] (2015). http://arxiv.org/abs/1508.01076
- 4.Aichholzer, O., Cheng, H., Devadoss, S.L., Hackl, T., Huber, S., Li, B., Risteski, A.: What makes a tree a straight skeleton? In: Proceedings of the 24th Canadian Conference on Computational Geometry, (CCCG 2012), pp. 253–258. Charlottetown, PE, Canada (2012)Google Scholar
- 5.Biedl, T., Held, M., Huber, S.: Recognizing straight skeletons and Voronoi diagrams and reconstructing their input. In: Gavrilova, M., Vyatkina, K. (eds.) Proceedings of the 10th International Symposium on Voronoi Diagrams in Science & Engineering (ISVD 2013), pp. 37–46. IEEE Computer Society, Saint Petersburg, Russia (2013)Google Scholar
- 6.Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane (Extended Abstract). In: Proceedings of 41st Annual ACM Symposium Theory Computing (STOC 2009), pp. 631–638. ACM, Bethesda, MD, USA (2009)Google Scholar