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Full-Text Articles in Mathematics
Unfolding Convex Polyhedra Via Quasigeodesic Star Unfoldings, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Unfolding Convex Polyhedra Via Quasigeodesic Star Unfoldings, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Computer Science: Faculty Publications
We extend the notion of a star unfolding to be based on a simple quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron P to a simple, planar polygon: shortest paths from all vertices of P to Q are cut, and all but one segment of Q is cut.
Singularities Of Hinge Structures, Ciprian Borcea, Ileana Streinu
Singularities Of Hinge Structures, Ciprian Borcea, Ileana Streinu
Computer Science: Faculty Publications
Motivated by the hinge structure present in protein chains and other molecular conformations, we study the singularities of certain maps associated to body-and-hinge and panel-and-hinge chains. These are sequentially articulated systems where two consecutive rigid pieces are connected by a hinge, that is, a codimension two axis. The singularities, or critical points, correspond to a dimensional drop in the linear span of the axes, regarded as points on a Grassmann variety in its Pl¨ucker embedding. These results are valid in arbitrary dimension. The three dimensional case is also relevant in robotics.
Enumerating Constrained Non-Crossing Minimally Rigid Frameworks, David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-Ichi Tanigawa
Enumerating Constrained Non-Crossing Minimally Rigid Frameworks, David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-Ichi Tanigawa
Computer Science: Faculty Publications
In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all the constrained non-crossing Laman …
Unfolding Manhattan Towers, Mirela Damian, Robin Flatland, Joseph O'Rourke
Unfolding Manhattan Towers, Mirela Damian, Robin Flatland, Joseph O'Rourke
Computer Science: Faculty Publications
We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a 4×5×1 refinement of the vertex grid.
Pebble Game Algorithms And Sparse Graphs, Audrey Lee, Ileana Streinu
Pebble Game Algorithms And Sparse Graphs, Audrey Lee, Ileana Streinu
Computer Science: Faculty Publications
A multi-graph G on n vertices is (k,ℓ)-sparse if every subset of n′⩽n vertices spans at most kn′-ℓ edges. G is tight if, in addition, it has exactly kn-ℓ edges. For integer valuesk and ℓ∈[0,2k), we characterize the (k,ℓ)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)-pebble games. [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs.
Cauchy’S Arm Lemma On A Growing Sphere, Zachary Abel, David Charlton, Sébastien Collette, Erik D. Demaine, Martin L. Demaine, Stefan Langerman, Joseph O'Rourke, Val Pinciu, Godfried Toussaint
Cauchy’S Arm Lemma On A Growing Sphere, Zachary Abel, David Charlton, Sébastien Collette, Erik D. Demaine, Martin L. Demaine, Stefan Langerman, Joseph O'Rourke, Val Pinciu, Godfried Toussaint
Computer Science: Faculty Publications
We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma.
Grid Vertex-Unfolding Orthogonal Polyhedra, Mirela Damian
Grid Vertex-Unfolding Orthogonal Polyhedra, Mirela Damian
Computer Science: Faculty Publications
No abstract provided.
A Class Of Convex Polyhedra With Few Edge Unfoldings, Alex Benton, Joseph O'Rourke
A Class Of Convex Polyhedra With Few Edge Unfoldings, Alex Benton, Joseph O'Rourke
Computer Science: Faculty Publications
We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does have (several) nonoverlapping edge unfoldings.