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4 added 670 characters in body

Represent the position of a unit-length, oriented segment $s$ in the plane by the location $a$ of its basepoint and an orientation $\theta$: $s = (a,\theta)$. So $s$ can be viewed 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 of unit-length segments as the average distance between their corresponding points:

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 distances between $(1-t) a_1 + t b_1$ and $(1-t) a_2 + t b_2$ as $t$ varies in $[0,1]$. This is indeed a metric, I believe, because the triangle inequality holds between corresponding points in three positions of the segment. This metric is intended to capture the intuitive notion of how much work 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 a point $p \in s$ should also be a geodesic, but even this is not so clear to me. Certainly a rotation about a point not on $s$ is (generally) not a geodesic. Of course the main interest would be in geodesics that mix translation and rotation, showing (locally) optimal repositioning 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 the Kakeya 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!

Addendum

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 Series in mathematics Mathematics and engineeringEngineering, 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 problem of S[.] Ulam..: Among all continuous motions of an oriented line segment $S$ in $\mathbb{E}^n$ from one position to another, which preserves its length [...], find one for which the sum of the lengths of the paths swept by its endpoints is minimal.

The concentration here on the motion of the endpoints—in contrast to the average distance metric I proposed—seems to render these results as not directly relevant, although nevertheless quite interesting.

3 "Ulam problem" reference.

Represent the position of a unit-length, oriented segment $s$ in the plane by the location $a$ of its basepoint and an orientation $\theta$: $s = (a,\theta)$. So $s$ can be viewed 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 of unit-length segments as the average distance between their corresponding points:

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 distances between $(1-t) a_1 + t b_1$ and $(1-t) a_2 + t b_2$ as $t$ varies in $[0,1]$. This is indeed a metric, I believe, because the triangle inequality holds between corresponding points in three positions of the segment. This metric is intended to capture the intuitive notion of how much work 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 a point $p \in s$ should also be a geodesic, but even this is not so clear to me. Certainly a rotation about a point not on $s$ is (generally) not a geodesic. Of course the main interest would be in geodesics that mix translation and rotation, showing (locally) optimal repositioning 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 the Kakeya 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!

Addendum. 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...

2 Added numerical distances.

Represent the position of a unit-length, oriented segment $s$ in the plane by the location $a$ of its basepoint and an orientation $\theta$: $s = (a,\theta)$. So $s$ can be viewed 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 of unit-length segments as the average distance between their corresponding points:

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 distances between $(1-t) a_1 + t b_1$ and $(1-t) a_2 + t b_2$ as $t$ varies in $[0,1]$. This is indeed a metric, I believe, because the triangle inequality holds between corresponding points in three positions of the segment. This metric is intended to capture the intuitive notion of how much work 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 a point $p \in s$ should also be a geodesic, but even this is not so clear to me. Certainly a rotation about a point not on $s$ is (generally) not a geodesic. Of course the main interest would be in geodesics that mix translation and rotation, showing (locally) optimal repositioning 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 the Kakeya 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!

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