Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Unfolding (4)
- Genus-zero (3)
- Grid unfolding (2)
- Orthogonal polyhedra (2)
- Polyhedron (2)
-
- Bar framework (1)
- Case studies (1)
- Collision-free motion (1)
- Computational geometry (1)
- Convex hull (1)
- Convex polyhedra (1)
- Data ethics (1)
- Delaunay complex (1)
- Development (1)
- Dissection (1)
- Education (1)
- Folding (1)
- General unfolding (1)
- Geometric flexibility (1)
- Geometric permutation (1)
- Geometry processing (1)
- Hinged dissection (1)
- Knots (1)
- Line transversal (1)
- Locked chains (1)
- Morphing (1)
- Orthogonal (1)
- Periodic framework (1)
- Polygonal chain (1)
- Polytope (1)
- Publication Year
Articles 1 - 30 of 64
Full-Text Articles in Mathematics
Integrating Data Science Ethics Into An Undergraduate Major, Benjamin Baumer, Randi L. Garcia, Albert Y. Kim, Katherine M. Kinnaird, Miles Q. Ott
Integrating Data Science Ethics Into An Undergraduate Major, Benjamin Baumer, Randi L. Garcia, Albert Y. Kim, Katherine M. Kinnaird, Miles Q. Ott
Statistical and Data Sciences: Faculty Publications
We present a programmatic approach to incorporating ethics into an undergraduate major in statistical and data sciences. We discuss departmental-level initiatives designed to meet the National Academy of Sciences recommendation for weaving ethics into the curriculum from top-to-bottom as our majors progress from our introductory courses to our senior capstone course, as well as from side-to-side through co-curricular programming. We also provide six examples of data science ethics modules used in five different courses at our liberal arts college, each focusing on a different ethical consideration. The modules are designed to be portable such that they can be flexibly incorporated …
Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, Joseph O'Rourke
Unfolding Convex Polyhedra Via Radially Monotone Cut Trees, Joseph O'Rourke
Computer Science: Faculty Publications
A notion of "radially monotone" cut paths is introduced as an effective choice for finding a non-overlapping edge-unfolding of a convex polyhedron. These paths have the property that the two sides of the cut avoid overlap locally as the cut is infinitesimally opened by the curvature at the vertices along the path. It is shown that a class of planar, triangulated convex domains always have a radially monotone spanning forest, a forest that can be found by an essentially greedy algorithm. This algorithm can be mimicked in 3D and applied to polyhedra inscribed in a sphere. Although the algorithm does …
Geometric Deformations Of Sodalite Frameworks, Ciprian Borcea, Ileana Streinu
Geometric Deformations Of Sodalite Frameworks, Ciprian Borcea, Ileana Streinu
Computer Science: Faculty Publications
In mathematical crystallography and computational materials science, it is important to infer flexibility properties of framework materials from their geometric representation. We study combinatorial, geometric and kinematic properties for frameworks modeled on sodalite.
Hypercube Unfoldings That Tile R3 And R2, Giovanna Diaz, Joseph O'Rourke
Hypercube Unfoldings That Tile R3 And R2, Giovanna Diaz, Joseph O'Rourke
Computer Science: Faculty Publications
We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an edge-unfolding that tiles the plane.
Spiral Unfoldings Of Convex Polyhedra, Joseph O'Rourke
Spiral Unfoldings Of Convex Polyhedra, Joseph O'Rourke
Computer Science: Faculty Publications
The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the polyhedra of revolution.
New And Improved Spanning Ratios For Yao Graphs, Luis Barba, Prosenjit Bose, Mirela Damian, Rolf Fagerberg, Wah Loon Keng, Joseph O'Rourke, André Van Renssen, Perouz Taslakian, Sander Verdonschot, Ge Xia
New And Improved Spanning Ratios For Yao Graphs, Luis Barba, Prosenjit Bose, Mirela Damian, Rolf Fagerberg, Wah Loon Keng, Joseph O'Rourke, André Van Renssen, Perouz Taslakian, Sander Verdonschot, Ge Xia
Computer Science: Faculty Publications
For a set of points in the plane and a fixed integer k > 0, the Yao graph Yk partitions the space around each point into k equiangular cones of angle Θ = 2π/k, and connects each point to a nearest neighbor in each cone. It is known for all Yao graphs, with the sole exception of Y5, whether or not they are geometric spanners. In this paper we close this gap by showing that for odd k ≥ 5, the spanning ratio of Yk is at most 1/(1−2sin(3Θ/8)), which gives the first constant upper bound for …
A 2-Chain Can Interlock With An Open 10-Chain, Bin Lu, Joseph O'Rourke, Jianyuan K. Zhong
A 2-Chain Can Interlock With An Open 10-Chain, Bin Lu, Joseph O'Rourke, Jianyuan K. Zhong
Computer Science: Faculty Publications
Abstract. It is an open problem, posed in [3], to determine the minimal k such that an open flexible k-chain can interlock with a flexible 2-chain. It was first established in [5] that there is an open 16-chain in a trapezoid frame that achieves interlocking. This was subsequently improved in [6] to establish interlocking between a 2-chain and an open 11-chain. Here we improve that result once more, establishing interlocking between a 2-chain and a 10-chain. We present arguments that indicate that 10 is likely the minimum.
Unfolding Prismatoids As Convex Patches: Counterexamples And Positive Results, Joseph O'Rourke
Unfolding Prismatoids As Convex Patches: Counterexamples And Positive Results, Joseph O'Rourke
Computer Science: Faculty Publications
We address the unsolved problem of unfolding prismatoids in a new context, viewing a “topless prismatoid” as a convex patch—a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for “petal unfolding” topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads …
Source Unfoldings Of Convex Polyhedra Via Certain Closed Curves, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Source Unfoldings Of Convex Polyhedra Via Certain Closed Curves, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Computer Science: Faculty Publications
Abstract. We extend the notion of a source unfolding of a convex polyhedron P to be based on a closed polygonal curve Q in a particular class rather than based on a point. The class requires that Q “lives on a cone” to both sides; it includes simple, closed quasigeodesics. Cutting a particular subset of the cut locus of Q (in P) leads to a non-overlapping unfolding of the polyhedron. This gives a new general method to unfold the surface of any convex polyhedron to a simple, planar polygon
Π/2-Angle Yao Graphs Are Spanners, Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'Rourke, Ben Seamone, Michiel Smid, Stefanie Wuhrer
Π/2-Angle Yao Graphs Are Spanners, Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'Rourke, Ben Seamone, Michiel Smid, Stefanie Wuhrer
Computer Science: Faculty Publications
We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23√ 2). Enroute to this, we also show that the Yao graph Y∞4 in the L∞ metric is a planar spanner with stretch factor 8.
Common Edge-Unzippings For Tetrahedra, Joseph O'Rourke
Common Edge-Unzippings For Tetrahedra, Joseph O'Rourke
Computer Science: Faculty Publications
It is shown that there are examples of distinct polyhedra, each with a Hamiltonian path of edges, which when cut, unfolds the surfaces to a common net. In particular, it is established for infinite classes of triples of tetrahedra.
Continuous Blooming Of Convex Polyhedra, Erik D. Demaine, Martin L. Demaine, Vi Hart, Joan Iacono, Stefan Langerman, Joseph O'Rourke
Continuous Blooming Of Convex Polyhedra, Erik D. Demaine, Martin L. Demaine, Vi Hart, Joan Iacono, Stefan Langerman, Joseph O'Rourke
Computer Science: Faculty Publications
We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.
Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu
Conical Existence Of Closed Curves On Convex Polyhedra, Joseph O'Rourke, Costin Vîlcu
Computer Science: Faculty Publications
Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support …
Convex Polyhedra Realizing Given Face Areas, Joseph O'Rourke
Convex Polyhedra Realizing Given Face Areas, Joseph O'Rourke
Computer Science: Faculty Publications
Given n ≥ 4 positive real numbers, we prove in this note that they are the face areas of a convex polyhedron if and only if the largest number is not more than the sum of the others.
A Note On Solid Coloring Of Pure Simplicial Complexes, Joseph O'Rourke
A Note On Solid Coloring Of Pure Simplicial Complexes, Joseph O'Rourke
Computer Science: Faculty Publications
We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in Rd may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R3 it says that tetrahedra in a collection may be "solid 4-colored" so that no two glued face-to-face receive the same color.
Flat Zipper-Unfolding Pairs For Platonic Solids, Joseph O'Rourke
Flat Zipper-Unfolding Pairs For Platonic Solids, Joseph O'Rourke
Computer Science: Faculty Publications
We show that four of the five Platonic solids' surfaces may be cut open with a Hamiltonian path along edges and unfolded to a polygonal net each of which can "zipper-refold" to a flat doubly covered parallelogram, forming a rather compact representation of the surface. Thus these regular polyhedra have particular flat "zipper pairs." No such zipper pair exists for a dodecahedron, whose Hamiltonian unfoldings are "zip-rigid." This report is primarily an inventory of the possibilities, and raises more questions than it answers.
On Folding A Polygon To A Polyhedron, Joseph O'Rourke
On Folding A Polygon To A Polyhedron, Joseph O'Rourke
Computer Science: Faculty Publications
We show that the open problem presented in "Geometric Folding Algorithms: Linkages, Origami, Polyhedra" [DO07] is solved by a theorem of Burago and Zalgaller [BZ96] from more than a decade earlier.
On Flat Polyhedra Deriving From Alexandrov's Theorem, Joseph O'Rourke
On Flat Polyhedra Deriving From Alexandrov's Theorem, Joseph O'Rourke
Computer Science: Faculty Publications
We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov's gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n3) time for polygons whose gluings are specified by n labels.
Star Unfolding Convex Polyhedra Via Quasigeodesic Loops, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Star Unfolding Convex Polyhedra Via Quasigeodesic Loops, Jin-Ichi Itoh, Joseph O'Rourke, Costin Vîlcu
Computer Science: Faculty Publications
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 ℘ toQ, and cut all but one segment of Q.
The Yao Graph Y6 Is A Spanner, Joseph O'Rourke
The Yao Graph Y6 Is A Spanner, Joseph O'Rourke
Computer Science: Faculty Publications
We prove that Y6 is a spanner. Y6 is the Yao graph on a set of planar points, which has an edge from each point x to a closest point y within each of the six angular cones of 60◦ surrounding x .
Highway Hull Revisited, Greg Aloupis, Jean Cardinal, Sébastien Collette, Ferran Hurtado, Stefan Langerman, Joseph O'Rourke, Belén Palop
Highway Hull Revisited, Greg Aloupis, Jean Cardinal, Sébastien Collette, Ferran Hurtado, Stefan Langerman, Joseph O'Rourke, Belén Palop
Computer Science: Faculty Publications
A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H. The highway hull H(S,H) of a point set S is the minimal set containing S as well as the shortest paths between all pairs of points in H(S,H), using the highway time distance. We provide a Θ(nlogn) worst-case …
Morphing Of Triangular Meshes In Shape Space, Stefanie Wuhrer, Prosenjit Bose, Chang Shu, Joseph O'Rourke, Alan Brunton
Morphing Of Triangular Meshes In Shape Space, Stefanie Wuhrer, Prosenjit Bose, Chang Shu, Joseph O'Rourke, Alan Brunton
Computer Science: Faculty Publications
We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space S. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in R3 . We then extend this shape space to arbitrary triangulations in 3D using a heuristic approach and show the practical use of the approach using experiments. Furthermore, we discuss a modified shape space that is useful for isometric skeleton morphing. All of the newly presented …
Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke
Some Properties Of Yao Y4 Subgraphs, Joseph O'Rourke
Computer Science: Faculty Publications
The Yao graph for k = 4, Y4, is naturally partitioned into four subgraphs, one per quadrant. We show that the subgraphs for one quadrant differ from the subgraphs for two adjacent quadrants in three properties: planarity, connectedness, and whether the directed graphs are spanners.
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.
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.
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.
Band Unfoldings And Prismatoids: A Counterexample, Joseph O'Rourke
Band Unfoldings And Prismatoids: A Counterexample, Joseph O'Rourke
Computer Science: Faculty Publications
This note shows that the hope expressed in [ADL+07]--that the new algorithm for edge-unfolding any polyhedral band without overlap might lead to an algorithm for unfolding any prismatoid without overlap--cannot be realized. A prismatoid is constructed whose sides constitute a nested polyhedral band, with the property that every placement of the prismatoid top face overlaps with the band unfolding.
Unfolding Restricted Convex Caps, Joseph O'Rourke
Unfolding Restricted Convex Caps, Joseph O'Rourke
Computer Science: Faculty Publications
This paper details an algorithm for unfolding a class of convex polyhedra, where each polyhedron in the class consists of a convex cap over a rectangular base, with several restrictions: the cap’s faces are quadrilaterals, with vertices over an underlying integer lattice, and such that the cap convexity is "radially monotone," a type of smoothness constraint. Extensions of Cauchy’s arm lemma are used in the proof of non-overlap.