## Distance

Let be a smooth curve in a Manifold from to with and . Then where is the Tangent Space of at . The Length of with respect to the Riemannian structure is given by

 (1)

and the distance between and is the shortest distance between and given by
 (2)

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

 (3)

where is a Binomial Coefficient. The distances between points are therefore subject to relationships, where
 (4)

For , 2, ..., this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... (Sloane's 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):

 (5)

This equation can be derived by writing
 (6)

and eliminating and from the equations for , , , , , and .

See also Arc Length, Cube Point Picking, Expansive, Length (Curve), Metric, Planar Distance, Point-Line Distance--2-D, Point-Line Distance--3-D, Point-Plane Distance, Point-Point Distance--1-D, Point-Point Distance--2-D, Point-Point Distance--3-D, Space Distance, Sphere

References

Gray, A. The Intuitive Idea of Distance on a Surface.'' §13.1 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 251-255, 1993.

Sloane, N. J. A. Sequence A000217/M2535 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972.