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Single-Vertex Origami and Spherical Expansive Motions

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Discrete and Computational Geometry (JCDCG 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3742))

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Abstract

We prove that all single-vertex origami shapes are reachable from the open flat state via simple, non-crossing motions. We also consider conical paper, where the total sum of the cone angles centered at the origami vertex is not 2π. For an angle sum less than 2π, the configuration space of origami shapes compatible with the given metric has two components, and within each component, a shape can always be reconfigured via simple (non-crossing) motions. Such a reconfiguration may not always be possible for an angle sum larger than 2π.

The proofs rely on natural extensions to the sphere of planar Euclidean rigidity results regarding the existence and combinatorial characterization of expansive motions. In particular, we extend the concept of a pseudo-triangulation from the Euclidean to the spherical case. As a consequence, we formulate a set of necessary conditions that must be satisfied by three-dimensional generalizations of pointed pseudo-triangulations.

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Author information

Authors and Affiliations

  1. Computer Science Department, Smith College, Northampton, MA, 01063, USA

    Ileana Streinu

  2. Department of Mathematics, York University, Toronto, M3J 1P3, Canada

    Walter Whiteley

Authors
  1. Ileana Streinu
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  2. Walter Whiteley
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Editor information

Editors and Affiliations

  1. Research Institute of Education, Tokai University, 2-28-4 Tomigaya, Shibuya-ku Tokio, Japan

    Jin Akiyama

  2. Ibaraki University, Nakanarusawa, 316-8511, Hitachi, Ibaraki, Japan

    Mikio Kano

  3. Tokai University, 317 Nishino, 410-0395, Numazu, Japan

    Xuehou Tan

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© 2005 Springer-Verlag Berlin Heidelberg

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Streinu, I., Whiteley, W. (2005). Single-Vertex Origami and Spherical Expansive Motions. In: Akiyama, J., Kano, M., Tan, X. (eds) Discrete and Computational Geometry. JCDCG 2004. Lecture Notes in Computer Science, vol 3742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11589440_17

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  • DOI: https://doi.org/10.1007/11589440_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30467-8

  • Online ISBN: 978-3-540-32089-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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