Straight skeleton

In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a polygon may involve parabolic curves.

Straight skeletons were first defined for simple polygons by Aichholzer et al.,[1] and generalized to planar straight line graphs by Aichholzer and Aurenhammer.[2]

Contents

Definition

The straight skeleton of a polygon is defined by a continuous shrinking process in which the edges of the polygon are moved inwards parallel to themselves at a constant speed. As the edges move in this way, the vertices where pairs of edges meet also move, at speeds that depend on the angle of the vertex. If one of these moving vertices collides with a nonadjacent edge, the polygon is split in two by the collision, and the process continues in each part. The straight skeleton is the set of curves traced out by the moving vertices in this process.

For example, part (i) of the illustration shows the straight skeleton of a polygon. Part (ii) shows the sequence of smaller polygons traced out during the shrinking process by the moving edges.

Algorithms

The straight skeleton may be computed by simulating the shrinking process by which it is defined; a number of variant algorithms for computing it have been proposed, differing in the assumptions they make on the input and in the data structures they use for detecting combinatorial changes in the input polygon as it shrinks.

Applications

Each point within the input polygon can be lifted into three-dimensional space by using the time at which the shrinking process reaches that point as the z-coordinate of the point. The resulting three-dimensional surface has constant height on the edges of the polygon, and rises at constant slope from them except for the points of the straight skeleton itself, where surface patches at different angles meet. In this way, the straight skeleton can be used as the set of ridge lines of a building roof, based on walls in the form of the initial polygon.[1][7] Part (iii) of the illustration depicts a surface formed from the straight skeleton in this way.

Demaine, Demaine and Lubiw used the straight skeleton as part of a technique for folding a sheet of paper so that a given polygon can be cut from it with a single straight cut, and related origami design problems.[8]

Barequet et al. use straight skeletons in an algorithm for finding a three-dimensional surface that interpolates between two given polygonal curves.[9]

Tănase and Veltkamp propose to decompose concave polygons into unions of convex regions using straight skeletons, as a preprocessing step for shape matching in image processing.[10]

Bagheri and Razzazi use straight skeletons to guide vertex placement in a graph drawing algorithm in which the graph drawing is constrained to lie inside a polygonal boundary.[11]

The straight skeleton can also be used to construct an offset curve of a polygon, with mitered corners, analogously to the construction of an offset curve with rounded corners formed from the medial axis.

As with other types of skeleton such as the medial axis, the straight skeleton can be used to collapse a two-dimensional area to a simplified one-dimensional representation of the area. For instance, Haunert and Sester describe an application of this type for straight skeletons in geographic information systems, in finding the centerlines of roads.[12][13]

Higher dimensions

Barequet et al. defined a version of straight skeletons for three-dimensional polyhedra, described algorithms for computing it, and analyzed its complexity on several different types of polyhedron.[14]

References

  1. ^ a b c Aichholzer, Oswin; Aurenhammer, Franz; Alberts, David; Gärtner, Bernd (1995). "A novel type of skeleton for polygons". Journal of Universal Computer Science 1 (12): 752–761. MR1392429. http://www.jucs.org/jucs_1_12/a_novel_type_of. 
  2. ^ a b Aichholzer, Oswin; Aurenhammer, Franz (1996). "Straight skeletons for general polygonal figures in the plane". Proc. 2nd Ann. Int. Conf. Computing and Combinatorics (COCOON '96). Lecture Notes in Computer Science, no. 1090, Springer-Verlag. pp. 117–126. http://www.igi.tugraz.at/auren/psfiles/aa-ssgpf-98.ps.gz. 
  3. ^ Huber, Stefan; Held, Martin (2010). "Computing straight skeletons of planar straight-line graphs based on motorcycle graphs". Proceedings of the 22nd Canadian Conference on Computational Geometry. http://www.cs.umanitoba.ca/~cccg2010/electronicProceedings/paper15.pdf.  Huber, Stefan; Held, Martin (2011). "Theoretical and pratical results on straight skeletons of planar straight-line graphs". Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (SCG'11), June 13–15, 2011, Paris, France. pp. 171–178. .
  4. ^ Eppstein, David; Erickson, Jeff (1999). "Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions". Discrete & Computational Geometry 22 (4): 569–592. doi:10.1007/PL00009479. MR1721026. http://compgeom.cs.uiuc.edu/~jeffe/pubs/pdf/cycles.pdf. 
  5. ^ Cheng, Siu-Wing; Vigneron, Antoine (2002). "Motorcycle graphs and straight skeletons". Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 156–165. http://www.cs.ust.hk/~scheng/pub/motorpub.pdf. 
  6. ^ Das, Gautam K.; Mukhopadhyay, Asish; Nandy, Subhas C.; Patil, Sangameswar; Rao, S. V. (2010). "Computing the straight skeletons of a monotone polygon in O(n log n) time". Proceedings of the 22nd Canadian Conference on Computational Geometry. http://www.cs.umanitoba.ca/~cccg2010/electronicProceedings/paper24.pdf. 
  7. ^ Bélanger, David (2000). "Designing Roofs of Buildings". http://www.sable.mcgill.ca/~dbelan2/roofs/roofs.html. 
  8. ^ Demaine, Erik D.; Demaine, Martin L.; Lubiw, Anna (1998). "Folding and cutting paper". Revised Papers from the Japan Conference on Discrete and Computational Geometry (JCDCG'98). Lecture Notes in Computer Science, no. 1763, Springer-Verlag. pp. 104–117. doi:10.1007/b75044. http://theory.lcs.mit.edu/~edemaine/papers/JCDCG98/. 
  9. ^ Barequet, Gill; Goodrich, Michael T.; Levi-Steiner, Aya; Steiner, Dvir (2003). "Straight-skeleton based contour interpolation". Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 119–127. http://www.cs.technion.ac.il/~barequet/teaching/cg/sp01/project/papers/Barequet-Goodrich.ps.gz. 
  10. ^ Tănase, Mirela; Veltkamp, Remco C. (2003). "Polygon decomposition based on the straight line skeleton". Proceedings of the 19th Annual ACM Symposium on Computational Geometry. pp. 58–67. doi:10.1145/777792.777802. 
  11. ^ Bagheri, Alireza; Razzazi, Mohammadreza (2004). "Drawing free trees inside simple polygons using polygon skeleton". Computing and Informatics 23 (3): 239–254. MR2165282. 
  12. ^ Haunert, Jan-Henrik; Sester, Monika (2008). "Area collapse and road centerlines based on straight skeletons". Geoinformatica 12 (2): 169–191. doi:10.1007/s10707-007-0028-x. 
  13. ^ Raleigh, David Baring (2008). Straight Skeleton Survey Adjustment Of Road Centerlines From Gps Coarse Acquisition Data: A Case Study In Bolivia. Masters thesis. Ohio State University, Geodetic Science and Surveying. http://etd.ohiolink.edu/view.cgi?acc_num=osu1221854081. 
  14. ^ Barequet, Gill; Eppstein, David; Goodrich, Michael T.; Vaxman, Amir (2008). "Straight skeletons of three-dimensional polyhedra". Proc. 16th European Symposium on Algorithms. Lecture Notes in Computer Science. 5193. Springer-Verlag. pp. 148–160. arXiv:0805.0022. doi:10.1007/978-3-540-87744-8_13. 

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