Abstract
An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a net, a connected planar piece with no overlaps. A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertex-unfolding allows faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of “gridding” of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n 2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3×1×1 refinement of the vertex grid.
Article PDF
Similar content being viewed by others
References
Biedl, T., Demaine, E., Demaine, M., Lubiw, A., O’Rourke, J., Overmars, M., Robbins, S., Whitesides, S.: Unfolding some classes of orthogonal polyhedra. In: Proc. 10th Canad. Conf. Comput. Geom., pp. 70–71, 1998
Damian, M., Flatland, R., O’Rourke, J.: Grid vertex-unfolding orthogonal polyhedra. In: 23rd Symp. Theoretical Aspects Comput. Sci., 2006. Lecture Notes Comput. Sci., vol. 3884, pp. 264–276. Springer, Berlin (2006)
Demaine, E.D., O’Rourke, J.: A survey of folding and unfolding in computational geometry. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. Cambridge University Press, Cambridge (2005)
Demaine, E.D., O’Rourke, J.: Open problems from CCCG 2004. In: Proc. 17th Canad. Conf. on Comput. Geom., pp. 303–306, 2005
Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007). http://www.gfalop.org
Demaine, E.D., Eppstein, D., Erickson, J., Hart, G.W., O’Rourke, J.: Vertex-unfoldings of simplicial manifolds. In: Bezdek, A. (ed.) Discrete Geometry, pp. 215–228. Dekker, New York (2003)
Demaine, E.D., Iacono, J., Langerman, S.: Grid vertex-unfolding of orthostacks. In: Japan Conf. Discrete Comput. Geom. 2004. Lecture Notes Comput. Sci., vol. 3742, pp. 76–82. Springer, Berlin (2005). Int. J. Comput. Geom. Appl. (to appear)
O’Rourke, J.: Folding and unfolding in computational geometry. In: Discrete Comput. Geom., Japan Conf. Discrete Comput. Geom., 1998. Lecture Notes Comput. Sci., vol. 1763, pp. 258–266. Springer, Berlin (2000).
Paterson, M.S., Yao, F.F.: Optimal binary space partitions for orthogonal objects. J. Algorithms 13, 99–113 (1992)
Schwartz, E.L., Shaw, A., Wolfson, E.: A numerical solution to the generalized map-maker’s problem: flattening nonconvex polyhedral surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 11(9), 1005–1008 (1989)
Tarini, M., Hormann, K., Cignoni, P., Montani, C.: Polycube-maps. ACM Trans. Graph. 23(3), 853–860 (2004)
Wang, C.-H.: Manufacturability-driven decomposition of sheet metal products. PhD thesis, Carnegie Mellon University, The Robotics Institute (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
This is a significant revision of the preliminary version that appeared in [2].
J. O’Rourke’s research was supported by NSF award DUE-0123154.
Rights and permissions
About this article
Cite this article
Damian, M., Flatland, R. & O’Rourke, J. Grid Vertex-Unfolding Orthogonal Polyhedra. Discrete Comput Geom 39, 213–238 (2008). https://doi.org/10.1007/s00454-007-9043-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-007-9043-9