文档介绍：Chapter 5 45
Direction
Consider the lines thru a particular vertex point. They must fill space by themselves
since any point will be on the ray determined by the vertex and itself. Each of the
rays issuing from the vertex will be distinct from all the others, having only the
vertex mon with the rest. How can this spray of rays, which constitutes a sort
of polar coordinate system, and their relationships be characterized?
For a vertex, if a particular one of its rays is picked out as a reference, then any other
ray thru that vertex will make an angle with that reference ray which will have some
derivatives. It will now be shown that, as it was previously suggested, the rays are
uniquely identified and ordered by these values in, at least, the non-degenerate case.
Clearly this could only be true within a plane. In more than two dimensions a planar
subspace would have to be considered. Here consequences of Axiom 7 will be used
for the first time.
Order Inequalities
Consider any three distinct rays, call them R, S and T, emanating from mon
vertex, V where S is in the interior of the angle (R,T). Then their mutual angle
derivatives can be related by various inequalities which will now be derived.
Drop segments from arbitrary fixed points A on R and B on S to a point, P, moving
about on T in the vicinity of the vertex. By the crossbar theorem there is a point Q
where the line containing AP intersects the line of direction S. These auxiliary
constructions, for both signs of VP , are shown in the figure below. For small VP
they create two small triangles, BQP and VQP, different from those which have been
considered before. These triangles collapse as VP goes to zero and so they are good
prospects for fruitful application of the triangle inequality. In this situation, the
Chapter 5 46
quantities PA and PB are always positive, QB is positive for sufficiently small VP
and both PQ and VQ have the same sign as VP .
R
A S
R