Tetrangle inequality

Easthope, P.L. and Havel, T.F. Computational experience with an algorithm for tetrangle inequality bound smoothing. Bull. Math. Biol. 1989, 53, 173-194. 

The tetrangle inequality is a consequence of the nonnegativity of the symmetric four-point Cayley-Menger determinants. These do not factorize like the three-point determinants, but they can be expanded as a quadratic function of one of the squared distances, e.g., D(3,4), as follows ... 

Figure 3 The upper (left) and lower (right) tetrangle inequality limits. A heavy solid line denotes the distance at its lower bound, while a light solid line denotes the distance at its upper bound a dashed line denotes the associated limit...

Figure 4 The Cauchy pentagon is a tensegrity framework, who.se struts (heavy lines) constitute lower bounds, and whose cables are upper bounds, on the associated distances. The upper limit on any one pair of nodes separated by a strut, or the lower limit on any one pair connected by a cable, are tetrangle inequality limits implied by the bounds that correspond to the remaining struts and cables. Such five-point tetrangle inequality limits would not be found by an algorithm that only iterates over all quadruples of atoms as above...

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