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
jens.wittenburg@kit.edu
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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