Isometrically deformable cones and cylinders carrying planar curves

TL;DR

This paper investigates isometric deformations of cones and cylinders with planar curves, providing solutions for discrete cones and linking them to Bricard octahedra, while highlighting open problems in the smooth case.

Contribution

It offers a closed-form solution for discrete isometric deformations of cones and connects these to Bricard octahedra, advancing geometric understanding.

Findings

Discrete cones correspond to caps of Bricard octahedra.

Closed form solutions are obtained for the discrete case.

The smooth case reduces to an open differential equation problem.

Abstract

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.