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) |

(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) |

(4) |

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) |

**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.

© 1996-9

1999-05-24