Fifth Property of the Euclidean Metric

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[[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]]
[[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]]
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For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>x_i</math> and <math>x_j</math> is defined
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For a list of points <math>\{x_\ell\in\mathbb{R}^n,\,\ell\!=\!1\ldots N\}</math> in Euclidean vector space, distance-square between points <math>\,x_i\,</math> and <math>\,x_j\,</math> is defined
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<math>\begin{array}{rl}d_{ij}
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<math>d_{ij}=||x_i-x_j||^2
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\!\!&=\,\|x_i-_{}x_j\|^2
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=(x_i-x_j)^{\rm T}(x_i-x_j)=||x_i||^2+||x_j||^2-2x^{\rm T}_ix_j\\\\
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~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^T_i\!x_j\\\\
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=\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}I&-I\\-I&I\end{array}\right]
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&=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{*{20}r}\!I&-I\\\!-I&I\end{array}\right]
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\left[\begin{array}{cc}x_i\\x_j\end{array}\right]</math>
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\left[\!\!\begin{array}{*{20}c}x_i\\x_j\end{array}\!\!\right]
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\end{array}</math>
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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 <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
namely, for Euclidean metric <math>\sqrt{d_{ij}}</math> in <math>\mathbb{R}^n</math>
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* <math>\sqrt{d_{ij}}\geq0\,,~~i\neq j</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''nonnegativity''')'''
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* <math>\sqrt{d_{ij}}\geq0\,,~~i\not= j</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''nonnegativity''')'''
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* <math>\sqrt{d_{ij}}=0\,,~~i=j</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''self-distance''')'''
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* <math>\sqrt{d_{ij}}=0~\Leftrightarrow~x_i=x_j</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''self-distance''')'''
* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''symmetry''')'''
* <math>\sqrt{d_{ij}}=\sqrt{d_{ji}}</math> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;'''('''symmetry''')'''
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* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> &nbsp;&nbsp;&nbsp;'''('''triangle inequality''')'''
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* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\not=\!j\!\not=\!k</math> &nbsp;&nbsp;&nbsp;'''('''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 <math>\mathbb{R}^n</math>,
Augmenting the four fundamental Euclidean metric properties in <math>\mathbb{R}^n</math>,
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&nbsp;for all &nbsp;<math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> ,
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&nbsp;for all &nbsp;<math>i_{},j_{},\ell\not= k_{}\!\in\!\{1\ldots_{}N\}</math> ,
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&nbsp;<math>i\!<\!j\!<\!\ell</math> , &nbsp;and for &nbsp;<math>N\!\geq_{\!}4</math>&nbsp; distinct points &nbsp;<math>\{x_k\}</math> , &nbsp;the inequalities
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&nbsp;<math>i\!<\!j\!<\!\ell</math> , &nbsp;and for &nbsp;<math>N\!\geq_{\!}4</math>&nbsp; distinct points &nbsp;<math>\,\{x_k\}\,</math> , &nbsp;the inequalities
<math>\begin{array}{cc}
<math>\begin{array}{cc}
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\end{array}</math>
\end{array}</math>
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where &nbsp;<math>\theta_{ikj}\!=_{}\!\theta_{jki}</math>&nbsp; is the angle between vectors at vertex &nbsp;<math>x_k</math> &nbsp;, &nbsp;&nbsp;must be satisfied at each point &nbsp;<math>x_k</math>&nbsp; regardless of affine dimension.
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where &nbsp;<math>\theta_{ikj}\!=_{}\!\theta_{jki}</math>&nbsp; is the angle between vectors at vertex &nbsp;<math>\,x_k\,</math>&nbsp;, &nbsp;must be satisfied at each point &nbsp;<math>\,x_k\,</math>&nbsp; regardless of affine dimension.
== References ==
== References ==
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* Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, 2007
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* Dattorro, [http://www.meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], Meboo, 2005

Current revision

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: 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.

References

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