Generic and Abstract Rigidity

Abstract

Rigidity We are all familiar with frameworks of rods attached at joints. A rod and joint frame-work gives rise to a simple mathematical model consisting of line segments in Euclidean 3-space with common endpoints. A deformation is a continuous one-parameter family of such frameworks. If a framework has only trivial deformations, e.g. translations and rotations, then it is said to be rigid. Before giving a more precise mathematical formulation, we can use simple geometry to explore these ideas. Consider the triangular prism of Figure la. It has an obvious deformation in which the bottom triangle is held fixed and the three posts rotate simultaneously about their lower endpoints, remaining parallel throughout. As the posts move, any two of them form the sides of a parallelogram, and so the triangles formed by their upper points are congruent, see Figure 1b. Another deformation is to keep the planes of the two triangles parallel and " screw down " the top triangle, as in Figure 1c. We can try to