Fifth property of the Euclidean metric
For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined
<math>\begin{array}{rl}d_{ij} \!\!&=\,\|x_i-_{}x_j\|^2 ~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\ &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right] \left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right] \end{array}</math>
Euclidean distance must satisfy the requirements imposed by any metric space:
- <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math> (nonnegativity)
- <math>\sqrt{d_{ij}}=0\,,~~i=j</math> (self-distance)
- <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> (symmetry)
- <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> (triangle inequality)
where <math>\sqrt{d_{ij}}</math> is the Euclidean metric in <math>\mathbb{R}^n</math>.
Fifth property of the Euclidean metric
(Relative-angle inequality.)
Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>, for all <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}_{\,}</math>, <math>i\!<\!j\!<\!\ell\,\,</math>, and for <math>N\!\geq_{\!}4</math> distinct points <math>\{x_k\}_{\,}</math>, the inequalities
<math>\begin{array}{cc} |\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\quad\quad&{\rm(a)}\\ \theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\quad\quad&{\rm(b)}\\ 0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi\quad\quad&{\rm(c)} \end{array}</math>
where <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math> is the angle between vectors at vertex <math>x_k</math>, must be satisfied at each point <math>x_k</math> regardless of affine dimension.
