Several definitions of a
geographical mean/center/midpoint are possible. The solution given here can be
described as the “center of gravity” of the given positions, and can be
compared to the centroid of a geometrical shape. Informally, the found mean
position can be described by the following: Assume a sphere that is a model of
the Earth, and let the sphere roll freely on a horizontal plane (in a uniform
gravitational field). Place weights at the given positions of the sphere, and
the weights will pull one side of the sphere down due to the gravitation (assuming
the positions are not antipodal). When the sphere is in equilibrium, the “center
of gravity”-position is the tangent point between the sphere and the plane.

(Another possible way to
define a midpoint is the position where the sum of surface distances (great
circle distances) from the original positions is at a minimum. This midpoint is
undefined when only two positions are given. If three positions are given in
one dimension as 0, 0, and 3, this midpoint is at 0, while the arithmetic mean
is 1. Iterations are probably needed to calculate this midpoint, and code for
this is not included at this web site.)