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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).

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
Full Text: DOI
[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.
[2] O. Aichholzer, M. Cetina, R. Fabila, J. Leaños, G. Salazar, J. Urrutia, Convexifying monotone polygons while maintaining internal visibility, manuscript, 2011. · Zbl 1375.68118
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[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.
[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
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