Continuously Flattening Polyhedra Using Straight Skeletons
Author(s)Demaine, Erik D.; Demaine, Martin L.; Itoh, Jin-ichi; Lubiw, Anna; Nara, Chie; O'Rourke, Joseph; Abel, Zachary Ryan; ... Show more Show less
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We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the flat folding, which answers an open question in the book Geometric Folding Algorithms. The primary creases in our folding process can be found in quadratic time, though necessarily, creases must roll continuously, and we show that the full crease pattern can be exponential in size. We show that our method solves the fold-and-cut problem for convex polyhedra in any dimension. As an additional application, we show how a limiting form of our algorithm gives a general design technique for flat origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress).
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Department of Mathematics
Proceedings of the Annual Symposium on Computational Geometry (SOCG '14)
Association for Computing Machinery (ACM)
Zachary Abel, Erik D. Demaine, Martin L. Demaine, Jin-ichi Itoh, Anna Lubiw, Chie Nara, and Joseph O'Rourke. 2014. Continuously Flattening Polyhedra Using Straight Skeletons. In Proceedings of the thirtieth annual symposium on Computational geometry (SOCG '14). ACM, New York, NY, USA, Pages 396, 10 pages.
Author's final manuscript