Abstract
We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron ℘ to a simple (nonoverlapping) planar polygon: cut along one shortest path from each vertex of ℘ to Q, and cut all but one segment of Q.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alexandrov, A.D.: Vypuklye Mnogogranniki. Gosudarstvennoe Izdatelstvo Tekhno-Teoreticheskoi Literatury, Moscow (1950). In Russian
Alexandrov, A.D.: Konvexe Polyeder. Akademie Verlag, Berlin (1958). Math. Lehrbucher und Monographien. Translation of the 1950 Russian edition
Alexandrov, A.D.: Convex Polyhedra. Springer, Berlin (2005). Monographs in Mathematics. Translation of the 1950 Russian edition by N.S. Dairbekov, S.S. Kutateladze, and A.B. Sossinsky
Alexandrov, A.D., Zalgaller, V.A.: Intrinsic Geometry of Surfaces. American Mathematical Society, Providence (1967)
Aronov, B., O’Rourke, J.: Nonoverlap of the star unfolding. Discrete Comput. Geom. 8, 219–250 (1992)
Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007). http://www.gfalop.org
Ieiri, K., Itoh, J.-i., Vîlcu, C.: Quasigeodesics and farthest points on convex surfaces. Submitted to Adv. Geom. (2010)
Itoh, J.-i., Vîlcu, C.: Criteria for farthest points on convex surfaces. Math. Nachr., To appear (2009)
Itoh, J.-i., Vîlcu, C.: Geodesic characterizations of isosceles tetrahedra. Preprint (2008)
Itoh, J.-i., O’Rourke, J., Vîlcu, C.: Unfolding convex polyhedra via quasigeodesics. Technical Report 085, Smith College, July (2007). arXiv:0707.4258v2 [cs.CG]
Itoh, J.-i., O’Rourke, J., Vîlcu, C.: Unfolding convex polyhedra via quasigeodesics: Abstract. In: Proc. 17th Annu. Fall Workshop Comput. Comb. Geom., November (2007)
Itoh, J.-i., O’Rourke, J., Vîlcu, C.: Source unfoldings of convex polyhedra with respect to certain closed polygonal curves. In: Proc. 25th European Workshop Comput. Geom., pp. 61–64. EuroCG, March (2009)
Kobayashi, S.: On conjugate and cut loci. In: Chern, S.S. (ed.) Studies in Global Geometry and Analysis, pp. 96–122. Mathematical Association of America, Washington (1967)
Miller, E., Pak, I.: Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings. Discrete Comput. Geom. 39, 339–388 (2008)
O’Rourke, J., Vîlcu, C.: A new proof for star unfoldings of convex polyhedra. Manuscript in preparation (2009)
Pogorelov, A.V.: Quasi-geodesic lines on a convex surface. Mat. Sb. 25(62), 275–306 (1949). English transl., Am. Math. Soc. Transl. 74 (1952)
Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. Translations of Mathematical Monographs, vol. 35. American Mathematical Society, Providence (1973)
Poincaré, H.: Sur les lignes géodésiques des surfaces convexes. Trans. Am. Math. Soc. 6, 237–274 (1905)
Sakai, T.: Riemannian Geometry. Translation of Mathematical Monographs, vol. 149. American Mathematical Society, Providence (1996)
Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. SIAM J. Comput. 15, 193–215 (1986)
Schreiber, Y., Sharir, M.: An optimal-time algorithm for shortest paths on a convex polytope in three dimensions. Discrete Comput. Geom. 39, 500–579 (2008)
Shiohama, K., Tanaka, M.: Cut loci and distance spheres on Alexandrov surfaces. Séminaires Congrès 1, 531–559 (1996). Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Itoh, Ji., O’Rourke, J. & Vîlcu, C. Star Unfolding Convex Polyhedra via Quasigeodesic Loops. Discrete Comput Geom 44, 35–54 (2010). https://doi.org/10.1007/s00454-009-9223-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9223-x