Edge-unfolding nested polyhedral bands

Greg Aloupis, Erik D. Demaine, Stefan Langerman, Pat Morin, Joseph O'Rourke, Ileana Streinu, Godfried Toussaint

    Research output: Contribution to journalArticle

    Abstract

    A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that all nested bands can be unfolded, by cutting along exactly one edge and folding continuously to place all faces of the band into a plane, without intersection.

    Original languageEnglish (US)
    Pages (from-to)30-42
    Number of pages13
    JournalComputational Geometry: Theory and Applications
    Volume39
    Issue number1
    DOIs
    StatePublished - Jan 1 2008

    Fingerprint

    Unfolding
    Intersection
    Polyhedron
    Convex polyhedron
    Convex polygon
    Orthogonal Projection
    Folding
    Polygon
    Face

    Keywords

    • Folding
    • Polyhedra
    • Slice curves

    ASJC Scopus subject areas

    • Computer Science Applications
    • Geometry and Topology
    • Control and Optimization
    • Computational Theory and Mathematics
    • Computational Mathematics

    Cite this

    Aloupis, G., Demaine, E. D., Langerman, S., Morin, P., O'Rourke, J., Streinu, I., & Toussaint, G. (2008). Edge-unfolding nested polyhedral bands. Computational Geometry: Theory and Applications, 39(1), 30-42. https://doi.org/10.1016/j.comgeo.2007.05.009

    Edge-unfolding nested polyhedral bands. / Aloupis, Greg; Demaine, Erik D.; Langerman, Stefan; Morin, Pat; O'Rourke, Joseph; Streinu, Ileana; Toussaint, Godfried.

    In: Computational Geometry: Theory and Applications, Vol. 39, No. 1, 01.01.2008, p. 30-42.

    Research output: Contribution to journalArticle

    Aloupis, G, Demaine, ED, Langerman, S, Morin, P, O'Rourke, J, Streinu, I & Toussaint, G 2008, 'Edge-unfolding nested polyhedral bands', Computational Geometry: Theory and Applications, vol. 39, no. 1, pp. 30-42. https://doi.org/10.1016/j.comgeo.2007.05.009
    Aloupis G, Demaine ED, Langerman S, Morin P, O'Rourke J, Streinu I et al. Edge-unfolding nested polyhedral bands. Computational Geometry: Theory and Applications. 2008 Jan 1;39(1):30-42. https://doi.org/10.1016/j.comgeo.2007.05.009
    Aloupis, Greg ; Demaine, Erik D. ; Langerman, Stefan ; Morin, Pat ; O'Rourke, Joseph ; Streinu, Ileana ; Toussaint, Godfried. / Edge-unfolding nested polyhedral bands. In: Computational Geometry: Theory and Applications. 2008 ; Vol. 39, No. 1. pp. 30-42.
    @article{1dbbc80ae5ee49e5a1f0222b48b53a83,
    title = "Edge-unfolding nested polyhedral bands",
    abstract = "A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that all nested bands can be unfolded, by cutting along exactly one edge and folding continuously to place all faces of the band into a plane, without intersection.",
    keywords = "Folding, Polyhedra, Slice curves",
    author = "Greg Aloupis and Demaine, {Erik D.} and Stefan Langerman and Pat Morin and Joseph O'Rourke and Ileana Streinu and Godfried Toussaint",
    year = "2008",
    month = "1",
    day = "1",
    doi = "10.1016/j.comgeo.2007.05.009",
    language = "English (US)",
    volume = "39",
    pages = "30--42",
    journal = "Computational Geometry: Theory and Applications",
    issn = "0925-7721",
    publisher = "Elsevier",
    number = "1",

    }

    TY - JOUR

    T1 - Edge-unfolding nested polyhedral bands

    AU - Aloupis, Greg

    AU - Demaine, Erik D.

    AU - Langerman, Stefan

    AU - Morin, Pat

    AU - O'Rourke, Joseph

    AU - Streinu, Ileana

    AU - Toussaint, Godfried

    PY - 2008/1/1

    Y1 - 2008/1/1

    N2 - A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that all nested bands can be unfolded, by cutting along exactly one edge and folding continuously to place all faces of the band into a plane, without intersection.

    AB - A band is the intersection of the surface of a convex polyhedron with the space between two parallel planes, as long as this space does not contain any vertices of the polyhedron. The intersection of the planes and the polyhedron produces two convex polygons. If one of these polygons contains the other in the projection orthogonal to the parallel planes, then the band is nested. We prove that all nested bands can be unfolded, by cutting along exactly one edge and folding continuously to place all faces of the band into a plane, without intersection.

    KW - Folding

    KW - Polyhedra

    KW - Slice curves

    UR - http://www.scopus.com/inward/record.url?scp=84867944320&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84867944320&partnerID=8YFLogxK

    U2 - 10.1016/j.comgeo.2007.05.009

    DO - 10.1016/j.comgeo.2007.05.009

    M3 - Article

    VL - 39

    SP - 30

    EP - 42

    JO - Computational Geometry: Theory and Applications

    JF - Computational Geometry: Theory and Applications

    SN - 0925-7721

    IS - 1

    ER -