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International Conference on Geometry and Graphics

ICGG 2022: ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics pp 115–126Cite as

Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type

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Part of the Lecture Notes on Data Engineering and Communications Technologies book series (LNDECT,volume 146)

Abstract

A. Kokotsakis studied the following problem in 1932: Given is a rigid closed polygonal line (planar or non-planar), which is surrounded by a polyhedral strip, where at each polygon vertex three faces meet. Determine the geometries of these closed strips with a continuous mobility. On the one side, we generalize this problem by allowing the faces, which are adjacent to polygon line-segments, to be skew; i.e. to be non-planar. But on the other side, we restrict to the case where the four angles associated with each polygon vertex fulfill the so-called isogonality condition that both pairs of opposite angles are equal or supplementary. In more detail, we study the case where the polygonal line is a skew quad, as this corresponds to a \((3\times 3)\) building block of a so-called V-hedra composed of skew quads. The latter also gives a positive answer to a question posed by R. Sauer in his book of 1970 whether continuous flexible skew quad surfaces exist.

Keywords

  • Kokotsakis belt
  • Continuous flexibility
  • Skew quad surfaces

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Notes

  1. 1.

    Assumed that this part of the continuous flexible polyhedra is not rigid.

  2. 2.

    Surfaces on which geodesic lines form a conjugate curve network [3].

  3. 3.

    This notation is in accordance with [6].

  4. 4.

    Note that in the remainder of the paper the indices are taken modulo n.

  5. 5.

    Corresponding faces and edges of these meshes are parallel.

  6. 6.

    The degree of the mobility corresponds to the number of rows plus columns of \(\mathscr {V}\) minus one (cf. Sauer [15, Theorem 11.18]).

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Acknowledgements

The research is supported by grant F77 (SFB “Advanced Computational Design”, subproject SP7) of the Austrian Science Fund FWF.

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry & Center for Geometry and Computational Design, TU Wien, Vienna, Austria

    Georg Nawratil

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  1. Georg Nawratil
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Correspondence to Georg Nawratil .

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  1. Department of Construction Engineering, University of São Paulo, São Paulo, Brazil

    Liang-Yee Cheng

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Nawratil, G. (2023). Generalizing Continuous Flexible Kokotsakis Belts of the Isogonal Type. In: Cheng, LY. (eds) ICGG 2022 - Proceedings of the 20th International Conference on Geometry and Graphics. ICGG 2022. Lecture Notes on Data Engineering and Communications Technologies, vol 146. Springer, Cham. https://doi.org/10.1007/978-3-031-13588-0_10

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  • DOI: https://doi.org/10.1007/978-3-031-13588-0_10

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