9867

Fermat-Weber Point of a Polygonal Chain

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.

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The origin of the Fermat–Weber problem is attributed to the great Pierre de Fermat (1601–1665) who asked Evangelista Torricelli (1608–1647) to find the point that minimizes the sum of the Euclidean distances to three given points , , in the plane. Around 1640, Torricelli devised a geometrical construction for this problem. He showed that subtends an angle of to each of the sides of the triangle . However, this is true only when each of the interior angles of the triangle is less than . The so-called complementary problem, where one angle of the triangle can be greater than first appeared in [1]. Its solution states that the optimum point always coincides with the obtuse vertex of the triangle; this was correctly proved later [2].
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)
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