Editing Distance geometry (section)

==Cayley–Menger determinants==
{{main|Cayley–Menger determinant}}
Cayley–Menger determinants, named after Arthur Cayley and Karl Menger, are determinants of matrices of distances between sets of points.

Let <math display="inline">A_0, A_1,\ldots, A_n</math> be ''n''&nbsp;+&nbsp;1 points in a semimetric space, their Cayley–Menger determinant is defined by

: <math>
\operatorname{CM}(A_0, \cdots, A_n) = \begin{vmatrix} 
0 & d_{01}^2 & d_{02}^2 & \cdots & d_{0n}^2 & 1 \\
d_{01}^2 & 0 & d_{12}^2 & \cdots & d_{1n}^2 & 1 \\
d_{02}^2 & d_{12}^2 & 0 & \cdots & d_{2n}^2 & 1 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
d_{0n}^2 & d_{1n}^2 & d_{2n}^2 & \cdots & 0 & 1 \\
1 & 1 & 1 & \cdots & 1 & 0
\end{vmatrix}</math>

If <math display="inline"> A_0, A_1,\ldots, A_n \in \mathbb R^k</math>, then they make up the vertices of a possibly [[Degeneracy (mathematics)|degenerate]] ''n''-simplex <math>v_n</math> in <math>\mathbb{R}^k</math>. It can be shown that<ref>{{Cite web|url=https://www.mathpages.com/home/kmath664/kmath664.htm|title=Simplex Volumes and the Cayley–Menger Determinant|website=www.mathpages.com|archive-url=https://web.archive.org/web/20190516033847/https://www.mathpages.com/home/kmath664/kmath664.htm|archive-date=16 May 2019|access-date=2019-06-08}}</ref> the ''n''-dimensional volume of the simplex <math>v_n</math> satisfies

: <math> \operatorname{Vol}_n(v_n)^2 = \frac{(-1)^{n+1}}{(n!)^2 2^n} \operatorname{CM}(A_0, \ldots, A_n). </math>

Note that, for the case of <math>n=0</math>, we have <math>\operatorname{Vol}_0(v_0) = 1</math>, meaning the "0-dimensional volume" of a 0-simplex is 1, that is, there is 1 point in a 0-simplex.

<math display="inline">A_0, A_1,\ldots, A_n</math> are affinely independent iff <math>\operatorname{Vol}_n(v_n) > 0</math>, that is, <math> (-1)^{n+1} \operatorname{CM}(A_0, \ldots, A_n) > 0</math>. Thus Cayley–Menger determinants give a computational way to prove affine independence.

If <math>
 k < n</math>, then the points must be affinely dependent, thus <math>
  \operatorname{CM}(A_0, \ldots, A_n) = 0</math>. Cayley's 1841 paper studied the special case of <math>
k = 3, n = 4</math>, that is, any five points <math>
A_0, \ldots, A_4</math> in 3-dimensional space must have <math>
\operatorname{CM}(A_0, \ldots, A_4) = 0</math>.