### 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.