#

#

# 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*(**R**^{n}),
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*(**R**^{n}) 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).

#