Fifth Property of the Euclidean Metric

(Difference between revisions)
 Revision as of 21:35, 4 December 2011 (edit)← Previous diff Revision as of 22:56, 22 September 2016 (edit) (undo)Next diff → 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 - $\begin{array}{rl}d_{ij} + [itex]d_{ij}=||x_i-x_j||^2 - \!\!&=\,\|x_i-_{}x_j\|^2 + =(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\ - ~=~(x_i-_{}x_j)^{\rm T}(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^{\rm T}_i\!x_j\\\\ + =\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right] - &=\,\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right] + \left[\begin{array}{cc}x_i\\x_j\end{array}\right]$ - \left[\!\!\begin{array}{cc}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]]

Revision as of 22:56, 22 September 2016

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 $LaTeX: \mathbb{R}^n$

• $LaTeX: \sqrt{d_{ij}}\geq0\,,~~i\neq j$                                       (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.