Fifth Property of the Euclidean Metric

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<math>\begin{array}{rl}d_{ij}
<math>\begin{array}{rl}d_{ij}
\!\!&=\,\|x_i-_{}x_j\|^2
\!\!&=\,\|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\\\\
+
~=~(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^T\quad x_j^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}</math>
\end{array}</math>

Revision as of 22:35, 4 December 2011

relative angle inequality tetrahedron
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)^{\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[\!\!\begin{array}{cc}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.

References

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