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].
To illustrate, the next applet
shows how the intersections M(s) (gray shaded regions) of the sets (A)s and (B)r-s 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 dilations 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).