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Journal for Geometry and Graphics 25 (2021), No. 2, 171--186
Copyright Heldermann Verlag 2021

Polyhedral Cylinders Formed by Kokotsakis Meshes

Jens Wittenburg
Institute for Technical Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany

Kokotsakis proved that an infinite planar mesh composed of congruent convex, non-trapezoidal, non-parallelogramic quadrilaterals is deformable with degree of freedom 1 in two modes if the quadrilaterals are rigid and if the edges are revolute joints. Stachel proved that in the deformed state the vertices of all quadrilaterals are located on a circular cylinder the radius of which is a free parameter. In other words: A Kokotsakis mesh forms two polyhedral cylinders which are deformable with degree of freedom one. Later, Stachel also investigated under which conditions a polyhedral cylinder is tiled by quadrilaterals. In the present paper new proofs and new results are obtained by using special parameters for quadrilaterals in combination with cylinder coordinates.

Keywords: Kokotsakis mesh, spherical four-bar, polyhedral cylinder, foldable and self-intersecting tiled polyhedral cylinder, periodic polyhedral cylinder, deltoid, parabola through four points.

MSC: 52C25; 51M20, 70B15.

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