Represent the position of a unit-length, oriented segment $s$ in the planeby the location $a$ of its basepoint andan orientation $\theta$: $s = (a,\theta)$. So $s$ can beviewed as a point in $\mathbb{R^2} \times \mathbb{S^1}$.Now I'll define a metric on this space.Define the distance $d(s_1,s_2)$ between two positions ofunit-length segments as the average distance between their correspondingpoints:
Above the distances are about 0.31, 0.61, and 0.53, left-to-right.
So if the endpoints of $s_i$ are $a_i$ and $b_i$,then $d(s_1,s_2)$ is the average of the Euclidean distancesbetween $(1-t) a_1 + t b_1$ and $(1-t) a_2 + t b_2$ as $t$ varies in $[0,1]$.This is indeed a metric, I believe, because the triangle inequalityholds between corresponding points in three positions of the segment.This metric is intended to capture the intuitive notion of how muchwork is required to move $s_1$ to $s_2$.
My question is: What are the geodesics in this space under this metric?Certainly a pure translation of $s$ is a geodesic.It seems that a pure rotation by at most $\pi$ of $s$ about apoint $p \in s$ should also be a geodesic, but even this isnot so clear to me. Certainly a rotation about a point not on $s$is (generally) not a geodesic. Of course the main interest would bein geodesics that mix translation and rotation, showing (locally) optimalrepositioning paths.
I investigated this long ago when working on motion-planning algorithms("moving a ladder"), but got quite blocked on this natural question.This superficially seems related to theKakeya needle problem,but the metric I propose does not measure swept area.Perhaps it has been studied in some guise previously.If so, a pointer would be appreciated. Thanks!
Addenda. (26Sep11.) I just ran across this book, by V. A. Dubovit͡s︡kiĭ,which seems relevant:The Ulam problem of optimal motion of line segments,Translation Series in Mathematics and Engineering, Optimization Software, 1985.It may take some time for me to locate a copy...
(11Nov11). I finally have this book in my hands.The Preface by Hestenes says,
Dubovitskij has succeeded in solving in closed form a generalization of a problemof S[.] Ulam..:Among all continuous motions of an oriented line segment $S$ in $\mathbb{E}^n$ from oneposition to another, which preserves its length [...], find one for which the sumof the lengths of the paths swept by its endpoints is minimal.
The concentration here on the motion of the endpoints—in contrast tothe average distance metric I proposed—seems to render these results as not directly relevant, although nevertheless quite interesting.