Algorithms for Designing Pop-Up Cards

Abel, Zachary; Demaine, Erik D.; Demaine, Martin L.; Eisenstat, Sarah; Lubiw, Anna; Schulz, André; Souvaine, Diane L.; Viglietta, Giovanni; Winslow, Andrew

doi:10.4230/LIPIcs.STACS.2013.269

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Algorithms for Designing Pop-Up Cards

Authors Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw, André Schulz, Diane L. Souvaine, Giovanni Viglietta, Andrew Winslow

Part of: Volume: 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)
Part of: Series: Leibniz International Proceedings in Informatics (LIPIcs)
Part of: Conference: Symposium on Theoretical Aspects of Computer Science (STACS)
License: Creative Commons Attribution-NoDerivs 3.0 Unported license
Publication Date: 2013-02-26

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LIPIcs.STACS.2013.269.pdf

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Document Identifiers

DOI: 10.4230/LIPIcs.STACS.2013.269
URN: urn:nbn:de:0030-drops-39407

Subject Classification

Keywords

geometric folding
linkages
universality

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Abstract

We prove that every simple polygon can be made as a (2D) pop-up card/book that opens to any desired angle between 0 and 360°.
More precisely, given a simple polygon attached to the two walls of the open pop-up, our polynomial-time algorithm subdivides the polygon into a single-degree-of-freedom linkage structure, such that closing the pop-up flattens the linkage without collision. This result solves an open problem of Hara and Sugihara from 2009. We also show how to obtain a more efficient construction for the special case of orthogonal polygons, and how to make 3D orthogonal polyhedra, from pop-ups that open to 90°, 180°, 270°, or 360°.

Cite As Get BibTex

Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw, André Schulz, Diane L. Souvaine, Giovanni Viglietta, and Andrew Winslow. Algorithms for Designing Pop-Up Cards. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 269-280, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013) https://doi.org/10.4230/LIPIcs.STACS.2013.269

Author Details

Zachary Abel

Erik D. Demaine

Martin L. Demaine

Sarah Eisenstat

Anna Lubiw

André Schulz

Diane L. Souvaine

Giovanni Viglietta

Andrew Winslow

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