Fermat-Weber Point of a Polygonal Chain

length of polygonal chain k

3

regular polygon of n sides

3

The Fermat–Weber point for a set of points

S

in the plane is the point

P

that minimizes the sum of the Euclidean distances from

P

to the points of

S

. A

k

-chain of a regular

n

-gon, denoted by

C

n

(k)

, is the segment of the boundary of the regular

n

-gon formed by a set of

k≤n

consecutive vertices of the regular

n

-gon. Here we consider chains with an odd number of vertices whose middle vertex is

(1,0)

. This Demonstration shows that for every fixed odd positive integer

k

, as

n

increases, the (blue) Fermat–Weber point of

C

n

(k)

moves on the

x

axis toward the point

(1,0)

on the boundary of the chain. It is also shown that when

n

exceeds a certain integer, say

N(k)

, the (red) Fermat–Weber point of

C

n

(k)

coincides with the vertex of the chain

(1,0)

lying on the

x

axis.