# Fifth Property of the Euclidean Metric

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## Revision as of 01:28, 10 January 2009

relative angle inequality tetrahedron

For a list of points $LaTeX: \{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}$ in Euclidean vector space, distance-square between points $LaTeX: \,x_i\,$ and $LaTeX: \,x_j\,$ is defined

$LaTeX: \begin{array}{rl}d_{ij} \!\!&=\,\|x_i-_{}x_j\|^2 ~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\ &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right] \left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right] \end{array}$

Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [Dattorro, ch.5.2]

namely, for Euclidean metric $LaTeX: \sqrt{d_{ij}}$ in $LaTeX: \mathbb{R}^n$

• $LaTeX: \sqrt{d_{ij}}\geq0\,,~~i\neq j$                                       (nonnegativity)
• $LaTeX: \sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j$                            (self-distance)
• $LaTeX: \sqrt{d_{ij}}=\sqrt{d_{ji}}$                                                  (symmetry)
• $LaTeX: \sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k$    (triangle inequality)

## Fifth property of the Euclidean metric (relative-angle inequality)

Augmenting the four fundamental Euclidean metric properties in $LaTeX: \mathbb{R}^n$,  for all  $LaTeX: i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}$ ,  $LaTeX: i\!<\!j\!<\!\ell$ ,  and for  $LaTeX: N\!\geq_{\!}4$  distinct points  $LaTeX: \,\{x_k\}\,$ ,  the inequalities

$LaTeX: \begin{array}{cc} |\theta_{ik\ell}-\theta_{\ell kj}|~\leq~\theta_{ikj\!}~\leq~\theta_{ik\ell}+\theta_{\ell kj}\\ \theta_{ik\ell}+\theta_{\ell kj}+\theta_{ikj\!}\,\leq\,2\pi\\ 0\leq\theta_{ik\ell\,},\theta_{\ell kj\,},\theta_{ikj}\leq\pi \end{array}$

where  $LaTeX: \theta_{ikj}\!=_{}\!\theta_{jki}$  is the angle between vectors at vertex  $LaTeX: \,x_k\,$ ,  must be satisfied at each point  $LaTeX: \,x_k\,$  regardless of affine dimension.