WOLFRAM|DEMONSTRATIONS PROJECT

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.