ABSTRACT
Consider a collection of n geometric objects, denoted by g1, g2,…, gn. These objects can be point-sets or tuples of point-sets. Assume that each gi has been dimensioned completely. Also assume that we are interested in dimensioning their relative positions so that we can create a rigid collection, that is, an n-tuple, of these n objects. In this collection there are nC2 pairwise relative positions that can be dimensioned. It turns out that graphs, described in some detail in Appendix 3, which should be read along with this chapter, are the best abstractions to capture such geometric relationships. Figure 8.1 illustrates this using a complete graph, where each node is a geometric object and each arc between two nodes stands for the relative position between the objects represented by the two nodes. (Note that each arc can contain from one up to six dimensions.) Chapter 7 showed how to dimension the relative position of any two of these arbitrary geometric objects. But do we need to dimension all of these relative positions to define the rigid collection, that is, an n-tuple, of these n objects? A little reflection indicates that the answer is no, because even for four coplanar points we don’t need 4C2=6 dimensions that would fix the relative positions within each pair of the four points-as we know, just 5 dimensions will do to specify a planar quadrilateral.
