The distance between two points is the length of the path connecting them. In the plane, the distance between points 
and 
is given by the Pythagorean
theorem,
 |
(1)
|
In Euclidean three-space, the distance between points 
and 
is
 |
(2)
|
In general, the distance between points 
and 
in a Euclidean space 
is given by
 |
(3)
|
For curved or more complicated surfaces, the so-called metric can be used to compute the distance between two points by integration. When unqualified, "the" distance generally means the shortest distance between two points. For example, there are an infinite number of paths between two points on a sphere but, in general, only a single shortest path. The shortest distance between two points is the length of a so-called geodesic between the points. In the case of the sphere, the geodesic is a segment of a great circle containing the two points.
Let 
be a smooth curve in a 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 shortest distance between 
and 
given by
 |
(5)
|
In order to specify the relative distances of 
points in the plane, 
coordinates are needed, since the first can always
be taken as (0, 0) and the second as 
, which defines the x-axis.
The remaining 
points need two coordinates each. However, the total number of distances is
 |
(6)
|
where 
is a binomial
coefficient. The distances between 
points are therefore subject to 
relationships, where
 |  |  |
(7)
|
 |  |  |
(8)
|
For 
, 2, ..., this gives 0, 0, 0, 1, 3,
6, 10, 15, 21, 28, ... (OEIS A000217) relationships,
and the number of relationships between 
points is the triangular
number 
.
Although there are no relationships for 
and 
points, for 
(a quadrilateral), there is one (Weinberg 1972):
 |  |