# Fifth Property of the Euclidean Metric

(Difference between revisions)
 Revision as of 02:28, 10 January 2009 (edit)m (Protected "Fifth Property of the Euclidean Metric" [edit=autoconfirmed:move=autoconfirmed])← Previous diff Current revision (00:00, 23 September 2016) (edit) (undo) (3 intermediate revisions not shown.) Line 2: Line 2: For a list of points $\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}$ in Euclidean vector space, distance-square between points $\,x_i\,$ and $\,x_j\,$ is defined For a list of points $\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}$ in Euclidean vector space, distance-square between points $\,x_i\,$ and $\,x_j\,$ is defined - $\begin{array}{rl}d_{ij} + [itex]d_{ij}=||x_i-x_j||^2 - \!\!&=\,\|x_i-_{}x_j\|^2 + =(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\ - ~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\ + =\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right] - &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right] + \left[\begin{array}{cc}x_i\\x_j\end{array}\right]$ - \left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right] + - \end{array}[/itex] + Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]] Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [[http://meboo.convexoptimization.com/BOOK/EuclideanDistanceMatrix.pdf Dattorro, ch.5.2]] namely, for Euclidean metric $\sqrt{d_{ij}}$ in $\mathbb{R}^n$ namely, for Euclidean metric $\sqrt{d_{ij}}$ in $\mathbb{R}^n$ - * $\sqrt{d_{ij}}\geq0\,,~~i\neq j$                                       '''('''nonnegativity''')''' + * $\sqrt{d_{ij}}\geq0\,,~~i\not= j$                                       '''('''nonnegativity''')''' * $\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j$                            '''('''self-distance''')''' * $\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j$                            '''('''self-distance''')''' * $\sqrt{d_{ij}}=\sqrt{d_{ji}}$                                                  '''('''symmetry''')''' * $\sqrt{d_{ij}}=\sqrt{d_{ji}}$                                                  '''('''symmetry''')''' - * $\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k$    '''('''triangle inequality''')''' + * $\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\not=\!j\!\not=\!k$    '''('''triangle inequality''')''' ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== ==Fifth property of the Euclidean metric '''('''relative-angle inequality''')'''== 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  $i_{},j_{},\ell\not= k_{}\!\in\!\{1\ldots_{}N\}$ ,  $i\!<\!j\!<\!\ell$ ,  and for  $N\!\geq_{\!}4$  distinct points  $\,\{x_k\}\,$ ,  the inequalities  $i\!<\!j\!<\!\ell$ ,  and for  $N\!\geq_{\!}4$  distinct points  $\,\{x_k\}\,$ ,  the inequalities

## Current revision

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 $LaTeX: d_{ij}=||x_i-x_j||^2 =(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\ =\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right] \left[\begin{array}{cc}x_i\\x_j\end{array}\right]$

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 $LaTeX: \mathbb{R}^n$

• $LaTeX: \sqrt{d_{ij}}\geq0\,,~~i\not= j$                                       (nonnegativity)
• $LaTeX: \sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j$                            (self-distance)
• $LaTeX: \sqrt{d_{ij}}=\sqrt{d_{ji}}$                                                  (symmetry)
• $LaTeX: \sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\not=\!j\!\not=\!k$    (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 $LaTeX: i_{},j_{},\ell\not= k_{}\!\in\!\{1\ldots_{}N\}$ , $LaTeX: i\!<\!j\!<\!\ell$ ,  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.