The Strange World
of the Hausdorff Metric Geometry
VIII. Intersections of Lines
Determined by One Point Sets
In the 2002 REU, we were also
able to determine the intersection properties of lines defined by single point
sets. In particular, we determined that distinct Hausdorff lines defined by
single point sets can intersect at infinitely many distinct elements, at only a
single point, and at no elements. The next two theorems (see Theorems 6 and 7 from
[7]) document
exactly when these various types of intersections occurred (at least in H(R2)). All of our results
extend to H(Rn),
but there is still one case to consider to complete the general classification.
The first theorem deals with the situation when we choose points from parallel
Euclidean lines.
Theorem 3: Let a ≠
b and f ≠ g be points in Rn. Let l(a, b)
denote the Euclidean line through a and b. Assume l(a, b)
is parallel to l(f,
g) and that a, b, f, g are coplanar. Let p be the point where the
perpendicular to l(f, g) through f
intersects l(a, b).
Let A = {a}, B = {b}, F = {f}, and G =
{g}. Then the Hausdorff lines L(A, B) and L(F,
G) have no
elements in H(Rn) in common if
and only if for some relabeling of a, b, f, and g we have p on
the Euclidean segment
defined by a and b.
Moreover, in all other cases L(A, B) and
L(F, G) have infinitely many elements in H(Rn) in common.
The following applet illustrates
this theorem. The points a, b, f, and g can be
moved to obtain different configurations. The points c0 (on
the Euclidean line through a and b) and c1 (on
the Euclidean line through f and g) as described in Theorem 2 can
also be moved. Experiment with different configurations to find elements in
common on the corresponding Hausdorff lines. Note that, according to Theorem 2,
any such element will need to be a subset of both (A)s and (B)t
for appropriate values of s and t and will also need to contain
both c0 and c1. When an appropriate
configuration is reached, you will notice that the intersection of (A)s
and (B)t is shaded.
Theorem 4 tells us what happens
when we choose points from intersecting Euclidean lines. The following applet
illustrates the situation. Again, the points a, b, f, g,
c0, and c1 can be moved to obtain different
configurations. The intersection point p is fixed.
Theorem 4: Let a ≠ b and f ≠ g be points in Rn so that l(a, b) ∩ l(f,
g) = p. Let A = {a}, B = {b},
F = {f}, and
G = {g}. Then L(A, B) ∩
L(F, G) = {p} if and only if for some
relabeling of a, b, f, and g we have
1. apb
and
2. either fpg or dE(g, p) ≤ dE(b, p) and dE(g, p) < dE(f, p).
Moreover, in all other cases L(A, B)
and L(F, G) have infinitely many elements in H(Rn) in common.