Distance
Let 
be a smooth curve in a Manifold 
from 
to 
with 
and 
. Then
where 
is the Tangent Space of at . The Lengthof 
with respect to the Riemannian structure is given by
 | (1) |
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 thefirst can always be taken as (0, 0) and the second as 
, which defines the x-Axis. Theremaining 
points need two coordinates each. However, the total number of distances is
 | (3) |
is a Binomial Coefficient. The distances between points are therefore subject to 
relationships,where | (4) |
, 2, ..., this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... (Sloane's A000217) relationships, and the number ofrelationships 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 
from the equations for 
, 
, 
, 
, 
, and 
.See also Arc Length, Cube Point Picking, Expansive, Length (Curve), Metric, PlanarDistance, Point-Line Distance--2-D, Point-Line Distance--3-D, Point-Plane Distance, Point-Point Distance--1-D, Point-PointDistance--2-D, Point-Point Distance--3-D, SpaceDistance, 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. SequenceA000217/M2535in ``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.
- Neumann Series (Integral Equation)
- Neusis Construction
- Neville's Algorithm
- Neville Theta Function
- Newcomb's Paradox
- Newman-Conway Sequence
- Newton-Cotes Formulas
- Newtonian Form
- Newton Number
- Newton-Raphson Fractal
- Newton-Raphson Method
- Newton's Backward Difference Formula
- Newton's Diverging Parabolas
- Newton's Divided Difference Interpolation Formula
- Newton's Formulas
- Newton's Forward Difference Formula
- Newton's Identities
- Newton's Iteration
- Newton's Method
- Newton's Parallelogram
- Newton's Relations
- Newton's Theorem
- Next Prime
- Nexus Number
- Neyman-Pearson Lemma