The Strange World of the Hausdorff Metric Geometry
X. Hausdorff Segments
We have seen that if A and
B are single point sets in H(Rn),
then there is always a point at each location between A and B. A
reasonable question to ask is if this is true in general. This question was
answered in [6]. In particular, if A and B are elements of H(Rn) and r = h(A,
B), then the set
M(s) = (A + s) ∩ (B + (r–s))
satisfies AM(s)B
with h(A, M(s)) = s for any s in [0, r].
In the 2003, 2004, and 2005 REUs,
we obtained many interesting results about betweenness in the Hausdorff metric
geometry. In 2003, I worked with Christopher Bay (then at Truman State
University, now at SUNY Stony Brook) and Amber Lembcke (Concordia College) to find
other sets C satisfying ACB. We also obtained some surprising
results about Hausdorff lines in general. In the 2004 REU, Kris Lund (then at GVSU, now at the
University of Nebraska-Lincoln), Patrick Sigmon (then at Wake Forest University,
now at North Carolina State University), and I obtained some additional results
on segments in H(Rn),
and found some connections to the Fibonacci and Lucas numbers. In 2005, Chantel
Blackburn (
The next theorem (Theorem 1 of [2])
shows that there may be many different elements at each location on a Hausdorff
segment.
Theorem 5: Let A and B be distinct elements of H(Rn) and let r
= h(A, B). Let s be a real number, 0 < s
< r, and let
t = r–s. If C is a compact subset of (A
+ s) ∩ (B + t) containing ∂[(A +
s) ∩ (B + t)], then C satisfies ACB.
In particular, this theorem is
more general than the result from [6]. To illustrate, the next applet shows how
the intersections (gray shaded regions) of the sets A+s and B+t
vary as s varies from 0 to h(A, B). The values of s
are represented by the position of the black point on the slider at the top
of the screen. Here s represents a specific location on the Hausdorff
segment between A and B. The values of s can be altered by
moving the black point on the slider. In this case, A is represented by
the red disk (think of A as the sun) and B by the blue crescent
(think of B as a new moon). The picture is not to scale. As the value of
s changes, the intersections of the extensions seem to “morph” the sun
into the moon, and vice-versa. I call this example the Unification Morph or,
alternatively, the Sun – Young Moon morph (see http://www.unification.net/ for information regarding the
nomenclature).
As we saw earlier, if A
and B are single point sets, then there is exactly one element C
in H(Rn) at
each location satisfying ACB. One consequence of Theorem 5 is that there
are many different possibilities sets containing more than one point. In the
Unification Morph we have an example of sets A and B for which we have
infinitely many different elements C
in H(Rn)
satisfying ACB at each location
between A and B. These are the two extremes. Under certain conditions, given by
Theorems 3.1-3.6 of [12] and summarized in the next theorem, it is
possible to have sets A and B so that the number of elements in H(Rn) satisfying ACB at each location between A and B is a finite number (greater than one). We call these conditions the PFAEL (Possibly
Finite At Each Location) conditions.
Theorem 6: Let A and B be distinct elements of H(Rn) and let r
= h(A, B). If there are finitely many elements C in
H(Rn)
satisfying ACB at each location, then
d(a, B) = r = d(b, A)
for every point a in A and
every point b in B.
Moreover, in these situations, the number of elements is
the same at every location.
One consequence of this theorem
is that if A and B are finite sets satisfying the PFAEL conditions, then the are
finitely many elements C in H(Rn) at each location satisfying ACB. In
the next section we will look at more examples in which there are finitely many
elements at each location between two elements A and B of H(Rn).
If either A or B is an infinite
set, the situation at first appears to be more complicated than in the finite
case. In Figures 2, 3, and 4, A is
the infinite set of points in red, B
the infinite set of points in blue, and C
the set of points in gray. In each case, C is the largest element at the indicated location satisfying ACB. In Figure 2, C is the only element at this location satisfying ACB. In Figure 3, we can delete any open
arc that is not the entire circle point from the inner of the two circles whose
union is C and obtain an element C* in H(Rn) satisfying AC*B at the same
location. So in this case, there are infinitely many elements at the indicated
location between A and B illustrating the fact that the PFAEL
conditions are necessary but not sufficient to ensure a finite number of
elements at each location between sets A
and B. In Figure 4, we can remove the
isolated point from C and obtain an
element C* in H(Rn) satisfying AC*B
at the same location. So with these infinite sets A and B of H(Rn), we can obtain 1,
2, or infinitely many elements C in H(Rn) satisfying ACB at a given location. However, as we
will see later, the infinite case is really no more complicated than the finite
case.
|
|
|
|
|
Figure 2 |
Figure 3 |
Figure 4 |
