×

Visibility-preserving convexifications using single-vertex moves. (English) Zbl 1237.68234

Summary: Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using single-vertex moves)? We prove the redundancy of the “single-vertex moves” condition: an affirmative answer to (1) implies an affirmative answer to (2). Since Aichholzer et al. recently proved (1), this settles (2).

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] O. Aichholzer, G. Aloupis, E.D. Demaine, M.L. Demaine, V. Dujmović, F. Hurtado, A. Lubiw, G. Rote, A. Schulz, D.L. Souvaine, A. Winslow, Convexifying polygons without losing visibilities, in: Proc. 23rd Annual Canadian Conference on Computational Geometry (CCCG) 2011, Toronto, Canada, pp. 229-234.; O. Aichholzer, G. Aloupis, E.D. Demaine, M.L. Demaine, V. Dujmović, F. Hurtado, A. Lubiw, G. Rote, A. Schulz, D.L. Souvaine, A. Winslow, Convexifying polygons without losing visibilities, in: Proc. 23rd Annual Canadian Conference on Computational Geometry (CCCG) 2011, Toronto, Canada, pp. 229-234.
[2] O. Aichholzer, M. Cetina, R. Fabila, J. Leaños, G. Salazar, J. Urrutia, Convexifying monotone polygons while maintaining internal visibility, manuscript, 2011.; O. Aichholzer, M. Cetina, R. Fabila, J. Leaños, G. Salazar, J. Urrutia, Convexifying monotone polygons while maintaining internal visibility, manuscript, 2011. · Zbl 1375.68118
[3] Connelly, R.; Demaine, E. D.; Rote, G., Straightening polygonal arcs and convexifying polygonal cycles, Discrete Comput. Geom., 30, 205-239 (2003) · Zbl 1046.52016
[4] E.D. Demaine, J. OʼRourke, Open problems from CCCG 2008, in: Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG2009), pp. 75-78.; E.D. Demaine, J. OʼRourke, Open problems from CCCG 2008, in: Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG2009), pp. 75-78.
[5] Devadoss, S. L.; Shah, R.; Shao, X.; Winston, E., Visibility graphs and deformations of associahedra · Zbl 1317.52018
[6] Streinu, I., Pseudo-triangulations, rigidity and motion planning, Discrete Comput. Geom., 34, 587-635 (2005) · Zbl 1084.68134
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.