Fermat-Weber Point of a Polygonal Chain

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The Fermat–Weber point for a set of points in the plane is the point
that minimizes the sum of the Euclidean distances from
to the points of
. A
-chain of a regular
-gon, denoted by
, is the segment of the boundary of the regular
-gon formed by a set of
consecutive vertices of the regular
-gon. Here we consider chains with an odd number of vertices whose middle vertex is
. This Demonstration shows that for every fixed odd positive integer
, as
increases, the (blue) Fermat–Weber point of
moves on the
axis toward the point
on the boundary of the chain. It is also shown that when
exceeds a certain integer, say
, the (red) Fermat–Weber point of
coincides with the vertex of the chain
lying on the
axis.
Contributed by: Bhaswar B. Bhattacharya (March 2011)
(Indian Statistical Institute, Kolkata, India)
Open content licensed under CC BY-NC-SA
Snapshots
Details
In this Demonstration we show how the position of the Fermat–Weber point for varies with
and
. It is observed that for every odd positive integer
, there exists an integer
such that whenever
, the Fermat–Weber point of
coincides with the vertex of
lying on the
axis. This can be thought of as an extension of Courant and Robbins' complementary problem on triangles as described above. The proof of this fact can be found in [3], where it is also shown that
, when
(
) is any odd positive integer.
The values of the quantities and
can be controlled using their respective sliders. For every fixed value of
, the control for the values of
can be set to autorun to observe the movement of the Fermat–Weber point of the chain, as described above. Note that the range of
in the Demonstration is set up to 21. The range can be increased by changing the range of the outer Manipulate function in the code.
References
[1] R. Courant and H. Robbins, What Is Mathematics?, New York: Oxford University Press, 1996.
[2] J. Krarup and S. Vajda, "On Torricelli's Geometrical Solution to a Problem of Fermat," IMA Journal of Mathematics Applied in Business and Industry, 8, 1997 pp. 215–224.
[3] B. B. Bhattacharya, "On the Fermat–Weber Point of a Polygonal Chain and Its Generalizations," Fundamenta Informaticae, to appear. (accepted October 2010)
Permanent Citation