We study the problem of shortest paths for a line segment in the plane
. As a measure of the distance traversed by a path, we take the averag
e curve length of the orbits of prescribed points on the line segment.
This problem is nontrivial even in free space (i.e., in the absence o
f obstacles). We characterize all shortest paths of the line segment m
oving in free space under the measure d2, the average orbit length of
the two endpoints. The problem of d2 optimal motion has been solved by
Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike
previous solutions, our basic tool is Cauchy's surface-area formula.
This new approach is relatively elementary, and yields new insights.