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On the Development of the Intersection of a Plane with a Polytope
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In this paper, the authors define a slice curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions.Abstract:
Define a ``slice'' curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex ``openings'' of a planar convex chain.read more
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Book ChapterDOI
Curves of Finite Total Curvature
TL;DR: In this article, the authors consider the class of curves of finite total curvature, as introduced by Milnor, and explore connections between discrete and differential geometry, and consider theorems of Fary/Milnor, Schur, Chakerian and Wienholtz.
Journal ArticleDOI
An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions
Yevgeny Schreiber,Micha Sharir +1 more
TL;DR: An optimal-time algorithm for computing the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions that constructs a dynamic version of Mount’s data structure that implicitly encodes the shortest paths from s to all other points of the surface.
Proceedings ArticleDOI
An optimal-time algorithm for shortest paths on a convex polytope in three dimensions
Yevgeny Schreiber,Micha Sharir +1 more
TL;DR: An optimal-time algorithm for computing the shortest-path map from a fixed source on the surface of a convex polytope in three dimensions that constructs a dynamic version of Mount's data structure that implicitly encodes the shortest paths from s to all other points of the surface.
Journal ArticleDOI
Edge-unfolding nested polyhedral bands
Greg Aloupis,Erik D. Demaine,Stefan Langerman,Pat Morin,Joseph O'Rourke,Ileana Streinu,Godfried T. Toussaint +6 more
TL;DR: It is proved that all nested bands can be unfolded, by cutting along exactly one edge and folding continuously to place all faces of the band into a plane, without intersection.
Reconfigurations of polygonal structures
TL;DR: In this paper, it was shown that polygonal structure reconfiguration can be performed in the dihedral model of motion, where angles and edge lengths are preserved, and the number of operations required to reconfigure between triangulations can be reduced using "point moves" and "edge flips".
References
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Book
Computational geometry in C
TL;DR: In this paper, the design and implementation of geometry algorithms arising in areas such as computer graphics, robotics, and engineering design are described and a self-contained treatment of the basic techniques used in computational geometry is presented.
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Proofs from The BOOK
TL;DR: Aigner and Ziegler as discussed by the authors present proofs for a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erds, including the spectral theorem from linear algebra, some more recent jewels like the Borromean rings and other surprises.
Book
Proofs from THE BOOK
Martin Aigner,Günter M. Ziegler +1 more
TL;DR: This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises.
Journal ArticleDOI
Rigidity and Energy
TL;DR: In this article, it was shown that if a rigid polygonal framework is obtained by reversing the roles of rods and cables, then it has a minimum energy function that minimizes the total energy in the framework.