The distance between two points in a metric space is the nonnegative real
number assigned to them by its metric (Lang 2013,
p. 1). In the plane, the distance between points 
and 
is given by the Pythagorean
theorem,
 |
(1)
|
In Euclidean space 
, the distance between points 
and 
is
 |
(2)
|
In general, the distance between points 
and 
in a Euclidean space 
is given by
 |
(3)
|
On a connected Riemannian manifold, the distance is the infimum of the lengths of joining piecewise smooth curves, and need not be attained by a shortest path. A shortest path, when it exists, can be parametrized as a geodesic. On a sphere, two distinct points that are not antipodal points are joined by a unique shortest path, the shorter arc of their great circle. Antipodal points on a sphere admit multiple shortest paths.
Let 
be a smooth
curve in a connected Riemannian
manifold 
from 
to 
with 
and 
. Then 
, where 
is the tangent space of
at 
. The curve length of 
with respect to the Riemannian
structure is given by
 |
(4)
|
and the distance 
between 
and 
is the infimum
of these lengths,
 |
(5)
|
Here the infimum is over joining piecewise smooth curves, which are smooth on each of finitely many subintervals (Lang 2013, p. 3).
For a generic configuration of 
points in the plane,
coordinates
specify their relative distances: the first point can be
placed at (0, 0) and the distinct second point at 
on the x-axis.
The remaining 
points need two coordinates
each. The total number of pairwise distances is
 |
(6)
|
where 
is a binomial
coefficient. Thus the generic number 
of independent algebraic relations among the pairwise distances
is
 |  |  |
(7)
|
 |  |  |
(8)
|
For 
there are no pairwise distances and
no such relations. For 
,
2, ..., the counts are therefore 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... (OEIS A000217).
For 
, the generic number of relations
is the triangular number 
.
There are no universal algebraic relations for 
or 
points, but for 
(a quadrilateral) there
is one (Weinberg 1972):
 |  |  |
(9)
|
This equation can be derived by writing
 |
(10)
|
and eliminating 
and 
from the equations for 
, 
,
, 
, 
,
and 
. This results in a Cayley-Menger
determinant
 |
(11)
|
as observed by Uspensky (1948, p. 256).
