The Geometry of Frameworks: Rigidity, Mechanisms and Cad

This paper is an introduction to the the study of constrained geometric structures. The motion and rigidity of frameworks of rods and joints is examined, and connections are drawn to other constrained structures that are useful in many applications, in particular to Computer Aided Design. 1. The geometry of frameworks Mathematical applications, beyond increasing our understanding of the world, often refocus our attention on the underlying mathematics, shifting our point of view and deepening our understanding of a familiar abstract relationship. The most mundane theorems may be transformed by this process of redirection and reinterpretation. Theorem 1 (SSS). If the lengths of corresponding sides of two triangles are equal, then the triangles are congruent. Let us consider this theorem in the context of two physical triangles constructed of straight rods joined at their endpoints, where the second triangle represents the result of forces and stresses acting on the first. We may interpret the theorem as stating that a triangular framework retains its structural integrity as long as the individual rods do, and as long as the joints don’t break apart. This means that a triangular framework made of rigid members is rigid even if each individual joint allows twisting, as does a pin joint, see Figure 1. There is no SSSS theorem for @ @ @ @ @ @ @ @ @ @ @ @ t t t @ t t d d@