(3.) ASME Y14.5M-1994 [Revision of ANSI Y14.5M-1992 (R1988)], Dimensioning
and Tolerancing, The American Society of Mechanical Engineers, New York, NY (1995).
The completeness property of proper dimensioning implies that it should be possible to tell the distance from any bar to each of the other bars of a properly dimensioned normalon.
Adopting a graph-theoretic approach, we define for each of the two bar-sets the following three alternative descriptions of a dimensioning graph.
[G.sub.s]([N.sub.r]), [G.sub.c]([N.sub.r]), and [G.sub.p]([N.sub.r]) are different representations of the same dimensioning graph of [N.sub.r], denoted G([N.sub.r].
The necessary and sufficient conditions that a normalon should satisfy in order for it to be properly dimensioned are stated in the normalon proper dimensioning theorem.
(Normalon Proper Dimensioning Theorem): Let [N.sub.r] be a dimensioned normalon or order r, let [G.sub.h]([N.sub.r]) and [G.sub.v]([N.sub.r]) be its horizontal and vertical dimensioning graphs, respectively.
First, given that the dimensioning is proper, we have to show that the condition is met.
To prove the first part of the two-way statement for condition (1), we assume that the normalon is properly dimensioned and need to prove that the dimensioning graph is a tree.
Since the assumption is that the dimensioning graph is a tree, it has no cycles, so the distance between any two bars can be calculated by one way at the most, which is following the only path between the two nodes in the tree representing the two bars between which the distance is sought.