We study cones and cylinders 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, which is used to prove that these cones correspond to caps of Bricard octahedra of the plane-symmetric type. For the smooth 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.

翻译：我们研究具有单参数等距变形且至少包含两条平面曲线的圆锥与圆柱，这些曲线在连续弯曲过程中保持平面性且位于非平行平面。该几何/运动学问题在光滑与离散两种框架下均被考察，因其是所谓T面拉链管推广构造的基础。与可轻松求解的圆柱情形不同，圆锥情形更为复杂，但我们成功给出了离散情形的闭式解，并据此证明此类圆锥对应于平面对称型Bricard八面体的帽盖。对于光滑情形，我们通过符号计算将问题简化为常微分方程，但其求解仍为未解难题。