Distance

Let $\gamma(t)$ be a smooth curve in a Manifold $M$ from $x$ to $y$ with $\gamma(0)=x$ and $\gamma(1)=y$. Then $\gamma'(t)\in T_{\gamma(t)},$ where $T_x$ is the Tangent Space of $M$ at $x$. The Length of $\gamma$ with respect to the Riemannian structure is given by

$$\int_0^1 \vert\vert\gamma'(t)\vert\vert _{\gamma(t)}\,dt,$$ (1)

and the distance $d(x,y)$ between $x$ and $y$ is the shortest distance between $x$ and $y$ given by

$$d(x,y)=\inf_{\gamma: x{\rm\ to\ }y} \int \vert\vert\gamma'(t)\vert\vert _{\gamma(t)}\,dt.$$ (2)

In order to specify the relative distances of $n>1$ points in the plane, $1+2(n-2) = 2n-3$ coordinates are needed, since the first can always be taken as (0, 0) and the second as $(x,0)$, which defines the x-Axis. The remaining $n-2$ points need two coordinates each. However, the total number of distances is

$${n\choose 2}={n!\over 2!(n-2)!}= {\textstyle{1\over 2}}n(n-1),$$ (3)

where ${n\choose k}$ is a Binomial Coefficient. The distances between $n>1$ points are therefore subject to $m$ relationships, where

$$m\equiv {\textstyle{1\over 2}}n(n-1)-(2n-3) = {\textstyle{1\over 2}}(n-2)(n-3).$$ (4)

For $n=1$, 2, ..., this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... (Sloane's A000217) relationships, and the number of relationships between $n$ points is the Triangular Number $T_{n-3}$.

Although there are no relationships for $n=2$ and $n=3$ points, for $n=4$ (a Quadrilateral), there is one (Weinberg 1972):

$$0 = d_{12}^4 d_{34}^2+d_{13}^4 d_{24}^2+d_{14}^4 d_{23}^2+d_{23}^4 d_{14}^2+d_{24}^4 d_{13}^2+d_{34}^4 d_{12}^2$$

$$\phantom{=} +d_{12}^2d_{23}^2d_{31}^2+d_{12}^2d_{24}^2d_{41}^2+d_{13}^2d_{34}^2d_{41}^2$$

$$\phantom{=} +d_{23}^2d_{34}^2d_{42}^2-d_{12}^2d_{23}^2d_{34}^2-d_{13}^2d_{32}^2d_{24}^2$$

$$\phantom{=} -d_{12}^2d_{24}^2d_{43}^2-d_{14}^2d_{42}^2d_{23}^2-d_{13}^2d_{34}^2d_{42}^2$$

$$\phantom{=} -d_{14}^2d_{43}^2d_{32}^2-d_{23}^2d_{31}^2d_{14}^2-d_{21}^2d_{13}^2d_{34}^2$$

$$\phantom{=} -d_{24}^2d_{41}^2d_{13}^2-d_{21}^2d_{14}^2d_{43}^2-d_{31}^2d_{12}^2d_{24}^2$$

$$\phantom{=} -d_{32}^2d_{21}^2d_{14}^2.$$ (5)

This equation can be derived by writing

$$d_{ij}\equiv\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}$$ (6)

and eliminating $x_i$ and $y_j$ from the equations for $d_{12}$, $d_{13}$, $d_{14}$, $d_{23}$, $d_{24}$, and $d_{34}$.

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.