342 research outputs found

    Locked and Unlocked Chains of Planar Shapes

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    We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

    Locked and unlocked smooth embeddings of surfaces

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    We study the continuous motion of smooth isometric embeddings of a planar surface in three-dimensional Euclidean space, and two related discrete analogues of these embeddings, polygonal embeddings and flat foldings without interior vertices, under continuous changes of the embedding or folding. We show that every star-shaped or spiral-shaped domain is unlocked: a continuous motion unfolds it to a flat embedding. However, disks with two holes can have locked embeddings that are topologically equivalent to a flat embedding but cannot reach a flat embedding by continuous motion.Comment: 8 pages, 8 figures. To appear in 34th Canadian Conference on Computational Geometr

    Force-Guiding Particle Chains for Shape-Shifting Displays

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    We present design and implementation of a chain of particles that can be programmed to fold the chain into a given curve. The particles guide an external force to fold, therefore the particles are simple and amenable for miniaturization. A chain can consist of a large number of such particles. Using multiple of these chains, a shape-shifting display can be constructed that folds its initially flat surface to approximate a given 3D shape that can be touched and modified by users, for example, enabling architects to interactively view, touch, and modify a 3D model of a building.Comment: 6 pages, 5 figure, submitted to IROS 201

    The GR2 gripper: an underactuated hand for open-loop in-hand planar manipulation

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    Performing dexterous manipulation of unknown objects with robot grippers without using high-fidelity contact sensors, active/sliding surfaces, or a priori workspace exploration is still an open problem in robot manipulation and a necessity for many robotics applications. In this paper, we present a two-fingered gripper topology that enables an enhanced predefined in-hand manipulation primitive controlled without knowing the size, shape, or other particularities of the grasped object. The in-hand manipulation behavior, namely, the planar manipulation of the grasped body, is predefined thanks to a simple hybrid low-level control scheme and has an increased range of motion due to the introduction of an elastic pivot joint between the two fingers. Experimental results with a prototype clearly show the advantages and benefits of the proposed concept. Given the generality of the topology and in-hand manipulation principle, researchers and designers working on multiple areas of robotics can benefit from the findings

    Fun with Fonts: Algorithmic Typography

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    Over the past decade, we have designed six typefaces based on mathematical theorems and open problems, specifically computational geometry. These typefaces expose the general public in a unique way to intriguing results and hard problems in hinged dissections, geometric tours, origami design, computer-aided glass design, physical simulation, and protein folding. In particular, most of these typefaces include puzzle fonts, where reading the intended message requires solving a series of puzzles which illustrate the challenge of the underlying algorithmic problem.Comment: 14 pages, 12 figures. Revised paper with new glass cane font. Original version in Proceedings of the 7th International Conference on Fun with Algorithm

    An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

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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 non-self-intersecting 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 that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application

    Complexity of Interlocking Polyominoes

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    Polyominoes are a subset of polygons which can be constructed from integer-length squares fused at their edges. A system of polygons P is interlocked if no subset of the polygons in P can be removed arbitrarily far away from the rest. It is already known that polyominoes with four or fewer squares cannot interlock. It is also known that determining the interlockedness of polyominoes with an arbitrary number of squares is PSPACE hard. Here, we prove that a system of polyominoes with five or fewer squares cannot interlock, and that determining interlockedness of a system of polyominoes including hexominoes (polyominoes with six squares) or larger polyominoes is PSPACE hard.Comment: 18 pages, 15 figure

    Tuning vortex fluctuations and the resistive transition in superconducting films with a thin overlayer

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    It is shown that the temperature of the resistive transition TrT_r of a superconducting film can be increased by a thin superconducting or normal overlayer. For instance, deposition of a highly conductive thin overlayer onto a dirty superconducting film can give rise to an "anti-proximity effect" which manifests itself in an initial increase of Tr(d2)T_r(d_2) with the overlayer thickness d2d_2 followed by a decrease of Tr(d2)T_r(d_2) at larger d2d_2. Such a nonmonotonic thickness dependence of Tr(d2)T_r(d_2) results from the interplay of the increase of a net superfluid density mitigating phase fluctuations and the suppression of the critical temperature TcT_c due to the conventional proximity effect. This behavior of Tr(d2)T_r(d_2) is obtained by solving the Usadel equations to calculate the temperature of the Berezinskii-Kosterletz-Thouless transition, and the temperature of the resistive transition due to thermally-activated hopping of single vortices in dirty bilayers. The theory incorporates relevant materials parameters such as thicknesses and conductivities of the layers, interface contact resistance between them and the subgap quasiparticle states which affect both phase fluctuations and the proximity effect suppression of TcT_c. The transition temperature TrT_r can be optimized by tuning the overlayer parameters, which can significantly weaken vortex fluctuations and nearly restore the mean-field critical temperature. The calculated behavior of Tr(d2)T_r(d_2) may explain the nonmonotonic dependence of Tr(d2)T_r(d_2) observed on (Ag, Au, Mg, Zn)-coated Bi films, Ag-coated Ga and Pb films or NbN and NbTiN films on AlN buffer layers. These results suggest that bilayers can be used as model systems for systematic investigations of optimization of fluctuations in superconductors
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