The Strange World of the Hausdorff Metric Geometry
XX. When Is
A Hausdorff Line Not A Line - 2?
If
either condition
d(A, B) > d(B, A) OR (A)s is a subset of Nr+s(B) for some s > 0
is not satisfied, then Theorems 6 and 7 of [2] show that we will have elements at every other location satisfying BAC or CBA. This ensures that when the conditions of (2) are satisfied, then our Hausdorff line will actually contain elements at every other location.
Theorem 8: Let A and B be elements of H(Rn) with r = d(B, A) ≥
d(A, B). For each s > 0 there is an element
C in H(Rn)
satisfying CAB with h(A, C) = s.
The set C described in the
previous theorem is constructed as follows: choose b0 in B
and a0 in A so that d(b0, a0)
= d(B, A). Then C = (A)s – Nr+s(b0)
satisfies CAB with h(A, C) = s.
Theorem 9: Let A and B be elements of H(Rn) with r = h(A, B). If
(A)s ∩
∂Nr+s(B) ≠ ø for some s > 0, then there
are infinitely many elements C in H(Rn) satisfying CAB
with h(A, C) = s.
In this case, choose x in (A)s ∩ ∂Nr+s(B).
Then for any q between
0 and s, let C be the union of (A)q with {x}. Then C satisfies CAB and h(A,
C) = s.
The next two applets illustrate
these theorems. The first expands on our previous two-point examples and uses
Theorem 8 to show elements at each location on the Hausdorff ray. The second
applet provides an example of a Hausdorff line for which there are elements at
every location on the line. In this example, A is a square and B
is a circle. Both Theorems 8 and 9 are used to construct points on this line.
The gray shaded regions are the elements at the given location on the ray or
line. The location of an element on the line is controlled by the black point
on the slider.
Hausdorff Rays
Hausdorff
Lines