The Strange World of the Hausdorff Metric Geometry
V. The Geometry of
the Hausdorff Metric:Circles
To build a geometry, we need a
collection of points along with lines that connect the points. In the
hyperspace H(Rn),
the points are elements of H(Rn),
namely the non-empty compact subsets of Rn.
So the geometry we will study is one in which the points are actual physical
objects, not the dimensionless points of Euclidean geometry. To build a
geometry, we need to define lines to connect these elements. In the 2000 GVSU
REU, Dominic Braun and I extended the standard Euclidean notions of circles and
lines to the space H(Rn).
We will begin with circles. A word on terminology: to distinguish between
“points” in the spaces Rn and H(Rn), we will use the word “point” to refer to a
point in Rn and the word “element” when referring to a
point in H(Rn).
Since we define a circle in
Euclidean space to be the set of points equidistant from a given point, why not
do the same in H(Rn)?
A circle in H(Rn)
centered at an element B in H(Rn) with radius r > 0 will be the set
of elements A in H(Rn)
so that h(B, A) = r. One of the results of the 2000
REU was that Dominic and I developed a complete characterization of circles in H(Rn), as described in
the following theorem (Theorem 2 from [7]) which can be formulated as follows:
Theorem 1 (The Circle
Theorem): Let A be an element of H(Rn)
and let r > 0. An element B in H(Rn)
satisfies h(B, A) = r if and only if
1. B
is a subset of
(A)r,
2. For each a in A, B intersects the closed ball of radius r centered at a, and
3. At
least one of the following is satisfied:
(a)
B ∩ ∂Nr(A) ≠ Ø, or
(b) there is a point a in A such that B ∩ ∂Nr(a) ≠ Ø and B ∩ Nr(a) = Ø.
To illustrate the Circle Theorem, let A be a two-point set {a1,
a2} (in red in the next
applet) and B the blue disk. Parts
(3a) and (3b) of the Circle Theorem show that B must contain a point on the boundary of one of Nr(a1) or Nr(a2) (drawn in red). This necessary
point appears in green. The user can move the green point along boundaries of Nr(a1) and Nr(a2), change the center of B, the location of the points a1 or a2, or the value of r
(via the black point). The program indicates the state (true/false) of each
of the conditions of the Circled Theorem for the given configuration. Play
around to see how B must be situated
to satisfy conditions 1, 2, and one of 3(a) or 3(b) and lie on the Hausdorff
Circle centered at A.
The characterization in the
Circle Theorem turns out to be very useful in proving facts about Hausdorff
lines as well as other properties of the space H(Rn) [15].
Dominic and I also began studying
lines in H(Rn). The next applet provides an illustration of
the different circles centered at the red rectangle A. Moving the black
point changes the radius, r, of the circle. Dominic and I proved that
every element in H(Rn)
that is r units from A must be a subset of (A)r. So while
there are many elements on the circle
centered at A of radius r, when drawn all together, they form (A)r.