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Synthesis of Fast and Collision-Free Folding of Polyhedral Nets
Synthesis of Fast and Collision-free Folding of Polyhedral Nets Yue Hao Yun-hyeong Kim Jyh-Ming Lien George Mason University Seoul National University George Mason University Fairfax, VA Seoul, South Korea Fairfax, VA [email protected] [email protected] [email protected] Figure 1: An optimized unfolding (top) created using our method and an arbitrary unfolding (bottom) for the fish mesh with 150 triangles (left). Each row shows the folding sequence by linearly interpolating the initial and target configurations. Self- intersecting faces, shown in red at bottom, result in failed folding. Additional results, foldable nets produced by the proposed method and an accompanied video are available on http://masc.cs.gmu.edu/wiki/LinearlyFoldableNets. ABSTRACT paper will provide a powerful tool to enable designers, materi- A predominant issue in the design and fabrication of highly non- als engineers, roboticists, to name just a few, to make physically convex polyhedral structures through self-folding, has been the conceivable structures through self-assembly by eliminating the collision of surfaces due to inadequate controls and the computa- common self-collision issue. It also simplifies the design of the tional complexity of folding-path planning. We propose a method control mechanisms when making deployable shape morphing de- that creates linearly foldable polyhedral nets, a kind of unfoldings vices. Additionally, our approach makes foldable papercraft more with linear collision-free folding paths. We combine the topolog- accessible to younger children and provides chances to enrich their ical and geometric features of polyhedral nets into a hypothesis education experiences. fitness function for a genetic-based unfolder and use it to mapthe polyhedral nets into a low dimensional space. -
© Cambridge University Press Cambridge
Cambridge University Press 978-0-521-85757-4 - Geometric Folding Algorithms: Linkages, Origami, Polyhedra Erik D. Demaine and Joseph O’Rourke Index More information Index 1-skeleton, 311, 339 bar, 9 3-Satisfiability, 217, 221 base, see origami, base, 2 α-cone canonical configuration, 151, 152 Bauhaus, 294 α-producible chain, 150, 151 Bellows theorem, 279, 348 δ-perturbation, 115 bending λ order function, 176, 177, 186 machine, xi, 13, 306 antisymmetry condition, 177, 186 pipe, 13, 14 consistency condition, 178, 186 sheet metal, 306 noncrossing condition, 179, 186 beta sheet, 158 time continuity, 174, 183, 187 Bezdek, Daniel, 331 transitivity condition, 178, 186 blooming, continuous, 333, 435 bond angle, 14, 131, 148, 151 Abe’s angle trisection, 286, 287 bond length, 148 accordion, 85, 193, 200, 261 active path, 244, 245, 247–249 cable, 53–55 acyclicity, 108 CAD, see cylindrical algebraic decomposition, 19 additor (Kempe), 32, 34, 35 cage, 21, 92, 93 Alexandrov, Aleksandr D., 348 canonical form, 74, 86, 87, 141, 151 Alexandrov’s theorem, 339, 348, 349, 352, 354, Cauchy’s arm lemma, 72, 133, 143, 145, 342, 343, 368, 381, 393, 419 377 existence, 351 Cauchy’s rigidity theorem, 43, 143, 213, 279, 339, uniqueness, 350 341, 342, 345, 348–350, 354, 403 algebraic motion, 107, 111 Cauchy—Steinitz lemma, 72, 342 algebraic set, 39, 44 chain algebraic variety, 27 4D, 92, 93, 437 alpha helix, 151, 157, 158 abstract, 65, 149, 153, 158 Amato, Nancy, 157 convex, 143, 145 amino acid, 158 cutting, xi, 91, 123 amino acid residue, 14, 148, 151 equilateral, see -
Marvelous Modular Origami
www.ATIBOOK.ir Marvelous Modular Origami www.ATIBOOK.ir Mukerji_book.indd 1 8/13/2010 4:44:46 PM Jasmine Dodecahedron 1 (top) and 3 (bottom). (See pages 50 and 54.) www.ATIBOOK.ir Mukerji_book.indd 2 8/13/2010 4:44:49 PM Marvelous Modular Origami Meenakshi Mukerji A K Peters, Ltd. Natick, Massachusetts www.ATIBOOK.ir Mukerji_book.indd 3 8/13/2010 4:44:49 PM Editorial, Sales, and Customer Service Office A K Peters, Ltd. 5 Commonwealth Road, Suite 2C Natick, MA 01760 www.akpeters.com Copyright © 2007 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photo- copying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Mukerji, Meenakshi, 1962– Marvelous modular origami / Meenakshi Mukerji. p. cm. Includes bibliographical references. ISBN 978-1-56881-316-5 (alk. paper) 1. Origami. I. Title. TT870.M82 2007 736΄.982--dc22 2006052457 ISBN-10 1-56881-316-3 Cover Photographs Front cover: Poinsettia Floral Ball. Back cover: Poinsettia Floral Ball (top) and Cosmos Ball Variation (bottom). Printed in India 14 13 12 11 10 10 9 8 7 6 5 4 3 2 www.ATIBOOK.ir Mukerji_book.indd 4 8/13/2010 4:44:50 PM To all who inspired me and to my parents www.ATIBOOK.ir Mukerji_book.indd 5 8/13/2010 4:44:50 PM www.ATIBOOK.ir Contents Preface ix Acknowledgments x Photo Credits x Platonic & Archimedean Solids xi Origami Basics xii -
On Rigid Origami I: Piecewise-Planar Paper with Straight-Line Creases
On Rigid Origami I: Piecewise-planar Paper with Straight-line Creases Zeyuan He, Simon D. Guest∗ January 5, 2021 Abstract We develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and results from cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions on important concepts in rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods, where the key results from each method are gathered and reviewed. Keywords: rigid-foldability, folding, configuration 1 Introduction This article develops a general theoretical framework for rigid origami, and uses this to gather and review the progress that researchers have made on the theory of rigid origami, including other related areas, such as rigidity theory, graph theory, linkage folding, and computer science. Origami has been used for many different physical models, as a recent review [1] shows. Sometimes a "rigid" origami model is required where all the deforma- tion is concentrated on the creases. A rigid origami model is usually considered to be a system of rigid panels that are able to rotate around their common boundaries and has been applied to many areas across different length scales [2]. These successful applications have inspired us to focus on the fundamental theory of rigid origami. Ultimately, we are considering two problems: first, the positive problem, which is to find useful sufficient and necessary conditions for a creased paper to be rigid-foldable; second, the inverse problem, which is to approximate a target surface by rigid origami. -
Arxiv:Math/0312253V1 [Math.MG] 12 Dec 2003 Contents ∗ † .Oe Rbesadcmlxt Sus41 35 Issues Complexity and Problems Open History 9
METRIC COMBINATORICS OF CONVEX POLYHEDRA: CUT LOCI AND NONOVERLAPPING UNFOLDINGS EZRA MILLER∗ AND IGOR PAK† 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 Rd, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in Rd 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 3-polytopes into R2. 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 non-polynomial complexity of nonconvex manifolds. Contents Introduction 2 Overview 2 Methods 4 1. Geodesics in polyhedral boundaries 7 2. Cut loci 10 3. Polyhedral nonoverlapping unfolding 15 4. The source poset 17 arXiv:math/0312253v1 [math.MG] 12 Dec 2003 5. -
View This Volume's Front and Back Matter
I: Mathematics Koryo Miura Toshikazu Kawasaki Tomohiro Tachi Ryuhei Uehara Robert J. Lang Patsy Wang-Iverson Editors http://dx.doi.org/10.1090/mbk/095.1 6 Origami I. Mathematics AMERICAN MATHEMATICAL SOCIETY 6 Origami I. Mathematics Proceedings of the Sixth International Meeting on Origami Science, Mathematics, and Education Koryo Miura Toshikazu Kawasaki Tomohiro Tachi Ryuhei Uehara Robert J. Lang Patsy Wang-Iverson Editors AMERICAN MATHEMATICAL SOCIETY 2010 Mathematics Subject Classification. Primary 00-XX, 01-XX, 51-XX, 52-XX, 53-XX, 68-XX, 70-XX, 74-XX, 92-XX, 97-XX, 00A99. Library of Congress Cataloging-in-Publication Data International Meeting of Origami Science, Mathematics, and Education (6th : 2014 : Tokyo, Japan) Origami6 / Koryo Miura [and five others], editors. volumes cm “International Conference on Origami Science and Technology . Tokyo, Japan . 2014”— Introduction. Includes bibliographical references and index. Contents: Part 1. Mathematics of origami—Part 2. Origami in technology, science, art, design, history, and education. ISBN 978-1-4704-1875-5 (alk. paper : v. 1)—ISBN 978-1-4704-1876-2 (alk. paper : v. 2) 1. Origami—Mathematics—Congresses. 2. Origami in education—Congresses. I. Miura, Koryo, 1930– editor. II. Title. QA491.I55 2014 736.982–dc23 2015027499 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Continuous Unfolding of Polyhedra – a Motion Planning Approach
Continuous Unfolding of Polyhedra – a Motion Planning Approach Zhonghua Xi and Jyh-Ming Lien Abstract— Cut along the surface of a polyhedron and unfold it foldability issue using motion planning techniques. First, we to a planar structure without overlapping is known as Unfolding propose a tessellation independent unfolding heuristic called Polyhedra problem which has been extensively studied in the Regularized Unfolding which generates identical unfoldings mathematics literature for centuries. However, whether there exists a continuous unfolding motion such that the polyhedron among polyhedra with different tessellations but who share can be continuously transformed to its unfolding has not been the same geometry. The proposed method allows unfoldings well studied. Recently, researchers started to recognize contin- to reuse unfolding motions. For a polyhedron with n faces, uous unfolding as a key step in designing and implementation its unfolding has n − 1 hingers which equals to the degree- of self-folding robots. In this paper, we model the unfolding of of-freedom (DOF) of the system or the dimensionality of a polyhedron as multi-link tree-structure articulated robot, and address this problem using motion planning techniques. Instead the configuration space. Planning motion in such high di- of sampling in continuous domain which traditional motion mensional space is nontrivial. We use the idea from [5], in planners do, we propose to sample only in the discrete domain. which instead of sampling in continuous domain, we sample Our experimental results show that sampling in discrete domain in the discrete domain. In our experiments, we show that is efficient and effective for finding feasible unfolding paths. -
BIOINSPIRED ORIGAMI: INFORMATION RETRIEVAL TECHNIQUES for DESIGN of FOLDABLE ENGINEERING APPLICATIONS a Dissertation by ELISSA M
BIOINSPIRED ORIGAMI: INFORMATION RETRIEVAL TECHNIQUES FOR DESIGN OF FOLDABLE ENGINEERING APPLICATIONS A Dissertation by ELISSA MORRIS Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, Daniel A. McAdams Committee Members, Richard Malak Douglas Allaire Michael Moreno Head of Department, Andres Polycarpou August 2019 Major Subject: Mechanical Engineering Copyright 2019 Elissa Morris ABSTRACT The science of folding has inspired and challenged scholars for decades. Origami, the art of folding paper, has led to the development of many foldable engineering solutions with applications in manufacturing, materials, and product design. Interestingly, three fundamental origami crease patterns are analogous to folding observed in nature. Numerous folding patterns, structures, and behaviors exist in nature that have not been considered for engineering solutions simply because they are not well-known or studied by designers. While research has shown applying biological solutions to engineering problems is significantly valuable, various challenges prevent the transfer of knowledge from biology to the engineering domain. One of those challenges is the retrieval of useful design inspiration. In this dissertation work, information retrieval techniques are employed to retrieve useful biological design solutions and a text-based search algorithm is developed to return passages where folding in nature is observed. The search -
Continuous Blooming of Convex Polyhedra
Noname manuscript No. (will be inserted by the editor) Continuous Blooming of Convex Polyhedra Erik D. Demaine · Martin L. Demaine · Vi Hart · John Iacono · Stefan Langerman · Joseph O'Rourke Abstract 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. 1 Introduction A standard approach to building 3D surfaces from rigid sheet material, such as sheet metal or cardboard, is to design an unfolding: cuts on the 3D surface so that the re- mainder can unfold (along edge hinges) into a single flat non-self-overlapping piece. The advantage of this approach is that many (relatively cheap) technologies|such as NC machining/milling, laser cutters, waterjet cutters, and sign cutters|enable man- ufacture of an arbitrary flat shape (with hinges) from a sheet of material. As a result, existence of and algorithms for unfolding have been studied extensively in the mathe- matical literature. Often overlooked in this literature, however, is the second manufacturing step: can we actually fold the flat shape along the hinges into the desired 3D surface, without E. D. Demaine and M. L. Demaine MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, fedemaine,[email protected] Partially supported by NSF CAREER award CCF-0347776. V. Hart http://vihart.com J. Iacono Department of Computer Science and Engineering, Polytechnic Institute of NYU, Brooklyn, NY, USA. http://john.poly.edu S. -
RIGID FLATTENING of POLYHEDRA with SLITS 1. Introduction in Many
RIGID FLATTENING OF POLYHEDRA WITH SLITS ZACHARY ABEL, ROBERT CONNELLY, ERIK D. DEMAINE, MARTIN L. DEMAINE, THOMAS C. HULL, ANNA LUBIW, AND TOMOHIRO TACHI 1. Introduction In many real-life situations we want polyhedra or polyhedral surfaces to flatten— think of paper bags, cardboard boxes, and foldable furniture. Although paper is flexible and can bend and curve, materials such as cardboard, metal, and plastic are not. The appropriate model for such non-flexible surfaces is “rigid origami” where the polyhedral faces are rigid and folding occurs only along pre-defined creases. In rigid origami, flattening is not always possible, and in fact, often no movement is possible at all. In particular, Cauchy’s theorem of 1813 says that if a convex polyhedron is made with rigid faces hinged at the edges then no movement is possible (see [Demaine and O’Rourke 07]). Connelly [Connelly 80] showed that this is true even if finitely many extra creases are added. However, cutting the surface of the polyhedron destroys rigidity and may even allow the polyhedron to be flattened. For example, a paper bag is a box whose top face has been removed, so the afore-mentioned rigidity results do not apply. Every- one knows the “standard” folds for flattening a paper bag. Surprisingly, these folds do not allow flattening with rigid faces unless the bag is short [Balkcom et al. 06]. Taller bags can indeed be flattened with rigid faces, but a different crease pattern is required [Wu and You 11]. Many of the clever ways of flattening cardboard boxes involve not only removal of the top face, but also extra slits and interlocking flaps in the bottom face. -
Continuous Blooming of Convex Polyhedra
Continuous Blooming of Convex Polyhedra The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Demaine, Erik D. et al. “Continuous Blooming of Convex Polyhedra.” Graphs and Combinatorics 27 (2011): 363-376. Web. 8 Dec. 2011. As Published http://dx.doi.org/10.1007/s00373-011-1024-3 Publisher Springer-Verlag Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/67481 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike 3.0 Detailed Terms http://creativecommons.org/licenses/by-nc-sa/3.0/ Noname manuscript No. (will be inserted by the editor) Continuous Blooming of Convex Polyhedra Erik D. Demaine · Martin L. Demaine · Vi Hart · John Iacono · Stefan Langerman · Joseph O'Rourke Abstract 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. 1 Introduction A standard approach to building 3D surfaces from rigid sheet material, such as sheet metal or cardboard, is to design an unfolding: cuts on the 3D surface so that the re- mainder can unfold (along edge hinges) into a single flat non-self-overlapping piece. The advantage of this approach is that many (relatively cheap) technologies|such as NC machining/milling, laser cutters, waterjet cutters, and sign cutters|enable man- ufacture of an arbitrary flat shape (with hinges) from a sheet of material. As a result, existence of and algorithms for unfolding have been studied extensively in the mathe- matical literature. -
On Rigid Origami I: Piecewise-Planar Paper with Straight-Line Creases
On Rigid Origami I: Piecewise-planar Paper with rspa.royalsocietypublishing.org Straight-line Creases Research Zeyuan He1, Simon D. Guest2 1;2Department of Engineering, University of Article submitted to journal Cambridge, Cambridge CB2 1PZ, United Kingdom Subject Areas: Origami (paper folding) is an effective tool for Structural Engineering transforming two-dimensional materials into three- dimensional structures, and has been widely applied Keywords: to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami rigid-foldability, generic where the facets are rigid, focusing on the kinematics rigid-foldability, folding, configuration of a panel-hinge model. Here we develop a theoretical framework for rigid origami, and show how this Author for correspondence: framework can be used to connect rigid origami Zeyuan He and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. e-mail: [email protected] First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analyzed using algebraic, geometric and numeric methods. In the algebraic part we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel- hinge system. In the geometric part we analyze corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.