# Fifth Property of the Euclidean Metric

(Difference between revisions)
 Revision as of 23:22, 31 October 2007 (edit)← Previous diff Revision as of 02:37, 9 November 2007 (edit) (undo) (→Fifth property of the Euclidean metric '''('''relative-angle inequality''')''')Next diff → Line 18: Line 18: ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== - + [[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]] Augmenting the four fundamental Euclidean metric properties in $\mathbb{R}^n$, Augmenting the four fundamental Euclidean metric properties in $\mathbb{R}^n$,  for all  $i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}$ ,  for all  $i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}$ ,

## Revision as of 02:37, 9 November 2007

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)
• $LaTeX: \sqrt{d_{ij}}=0\,,~~i=j$                                       (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)

relative angle inequality tetrahedron

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.