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27
Folding and Unfolding in Computational Geometry
"... Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain ..."
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Cited by 57 (4 self)
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Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?
Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings
, 2003
"... Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by id ..."
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Cited by 20 (6 self)
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Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v ∈ S, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3polytopes into R 2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic nonpolynomial complexity of nonconvex manifolds.
Folding and Unfolding
 in Computational Geometry. 2004. Monograph in preparation
, 2001
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, ..."
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Cited by 16 (3 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, and for that wonderful experience I owe many thanks. I had two excellent advisors, Anna Lubiw and Ian Munro. I started working with Anna after I took her two classes on algorithms and computational geometry during my Master’s, which got me excited about both these areas, and even caused me to switch entire fields of computer science, from distributed systems to theory and algorithms. Anna introduced me to Ian when some of our problems in computational geometry turned out to have large data structural components, and my work with Ian blossomed from there. The sets of problems I worked on with Anna and Ian diverged, and both remain my primary interests. Anna and Ian have had a profound influence throughout my academic career. At the most
VertexUnfoldings of Simplicial Manifolds
"... We present an algorithm to unfold any triangulated 2manifold (in particular, any simplicial polyhedron) into a nonoverlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles ..."
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Cited by 15 (3 self)
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We present an algorithm to unfold any triangulated 2manifold (in particular, any simplicial polyhedron) into a nonoverlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.
Reconfigurations of polygonal structures
, 2005
"... This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygona ..."
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Cited by 11 (1 self)
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This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of noncrossing straight edges joining them. A polyhedral surface is a threedimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a welldefined way of transforming one instance of a structure to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unitlength edges and angles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal
When Can a Net Fold to a Polyhedron?
 In Proceedings of the 11th Canadian Conference on Computational Geometry
, 1999
"... this paper, we study the problem of whether a polyhedron can be obtained from a net , i.e., a polygon and a set of creases, by folding along the creases. We consider two cases, depending on whether we are given the dihedral angle at each crease. If these dihedral angles are given the problem can be ..."
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Cited by 8 (1 self)
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this paper, we study the problem of whether a polyhedron can be obtained from a net , i.e., a polygon and a set of creases, by folding along the creases. We consider two cases, depending on whether we are given the dihedral angle at each crease. If these dihedral angles are given the problem can be solved in polynomial time by the simple expedient of performing the folding. If the dihedral angles are not given the problem is NPcomplete, at least for orthogonal polyhedra. We then turn to the actual folding process, and show an example of a net with rigid faces that can, in the sense above, be folded to form an orthogonal polyhedron, but only by allowing faces to intersect each other during the folding process.
An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures
, 2002
"... We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple nonselfintersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several ye ..."
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Cited by 5 (1 self)
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We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple nonselfintersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints...
Optimized Topological Surgery for Unfolding 3D Meshes
"... Constructing a 3D papercraft model from its unfolding has been fun for both children and adults since we can reproduce virtual 3D models in the real world. However, facilitating the papercraft construction process is still a challenging problem, especially when the shape of the input model is comple ..."
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Cited by 3 (0 self)
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Constructing a 3D papercraft model from its unfolding has been fun for both children and adults since we can reproduce virtual 3D models in the real world. However, facilitating the papercraft construction process is still a challenging problem, especially when the shape of the input model is complex in the sense that it has large variation in its surface curvature. This paper presents a new heuristic approach to unfolding 3D triangular meshes without any shape distortions, so that we can construct the 3D papercraft models through simple atomic operations for gluing boundary edges around the 2D unfoldings. Our approach is inspired by the concept of topological surgery, where the appearance of boundary edges of the unfolded closed surface can be encoded using a symbolic representation. To fully simplify the papercraft construction process, we developed a geneticbased algorithm for unfolding the 3D mesh into a single connected patch in general, while optimizing the usage of the paper sheet and balance in the shape of that patch. Several examples together with user studies are included to demonstrate that the proposed approach works well for a broad range of 3D triangular meshes. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems mesh unfolding, topological surgery, genetic algorithms, papercraft models 1.
Tighter Bounds on the Genus of Nonorthogonal Polyhedra Built from Rectangles
"... We prove that there is a polyhedron with genus 6 whose faces are orthogonal polygons (equivalently, rectangles) and yet the angles between some faces are not multiples , so the polyhedron itself is not orthogonal. On the other hand, we prove that any such polyhedron must have genus at least 3. The ..."
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Cited by 3 (3 self)
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We prove that there is a polyhedron with genus 6 whose faces are orthogonal polygons (equivalently, rectangles) and yet the angles between some faces are not multiples , so the polyhedron itself is not orthogonal. On the other hand, we prove that any such polyhedron must have genus at least 3. These results improve the bounds of Donoso and O'Rourke [4] that there are nonorthogonal polyhedra with orthogonal faces and genus 7 or larger, and any such polyhedron must have genus at least 2. We also demonstrate nonoverlapping onepiece edgeunfoldings (nets) for the genus7 and genus6 polyhedra.