On the flexibility of Kokotsakis meshes

TL;DR

This paper investigates the geometric and algebraic properties of Kokotsakis meshes, focusing on their flexibility and infinitesimal flexibility, and provides new conditions and equations characterizing their flexibility.

Contribution

It introduces a novel approach to analyze Kokotsakis meshes' flexibility using classical geometric theorems and derives explicit conditions for 3x3 quadrilateral meshes.

Findings

Infinitesimal flexibility conditions expressed via Ceva and Menelaus theorems.

Semi-algebraic description of the set of flexible meshes.

Flexibility conditions for 3x3 meshes in terms of face angles.

Abstract

In this paper we study geometric, algebraic, and computational aspects of flexibility and infinitesimal flexibility of Kokotsakis meshes. A Kokotsakis mesh is a mesh that consists of a face in the middle and a certain band of faces attached to the middle face by its perimeter. In particular any 3x3-mesh made of quadrangles is a Kokotsakis mesh. We express the infinitesimal flexibility condition in terms of Ceva and Menelaus theorems. Further we study semi-algebraic properties of the set of flexible meshes and give equations describing it. For 3x3-meshes we obtain flexibility conditions in terms of face angles.