Isometrically Deformable Cones and Cylinders Carrying Planar Curves

Abstract

We study conical and cylindrical surfaces with a 1-parametric isometric deformation carrying at least two planar curves, which remain planar during this continuous flexion and are located in non-parallel planes. We investigate this geometric/kinematic problem in the smooth and the discrete setting, as it is the base for a generalized construction of so-called T-hedral zipper tubes. In contrast to the cylindrical case, which can be solved easily, the conical one is more tricky, but we succeed to give a closed form solution for the discrete case. This is used to prove that these cones correspond to caps of Bricard octahedra of the plane-symmetric type. For the smooth conical case we are able to reduce the problem by means of symbolic computation to an ordinary differential equation, but its solution remains an open problem.