Fifth Property of the Euclidean Metric
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(Difference between revisions)
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[[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]] | [[Image:Thefifth.jpg|thumb|right|260px|relative angle inequality tetrahedron]] | ||
- | 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 | + | 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 |
<math>\begin{array}{rl}d_{ij} | <math>\begin{array}{rl}d_{ij} | ||
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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>, | ||
for all <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> , | for all <math>i_{},j_{},\ell\neq k_{}\!\in\!\{1\ldots_{}N\}</math> , | ||
- | <math>i\!<\!j\!<\!\ell</math> , and for <math>N\!\geq_{\!}4</math> distinct points <math>\{x_k\}</math> , the inequalities | + | <math>i\!<\!j\!<\!\ell</math> , and for <math>N\!\geq_{\!}4</math> distinct points <math>\,\{x_k\}\,</math> , the inequalities |
<math>\begin{array}{cc} | <math>\begin{array}{cc} | ||
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\end{array}</math> | \end{array}</math> | ||
- | where <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math> is the angle between vectors at vertex <math>x_k</math> , | + | where <math>\theta_{ikj}\!=_{}\!\theta_{jki}</math> is the angle between vectors at vertex <math>\,x_k\,</math> , must be satisfied at each point <math>\,x_k\,</math> regardless of affine dimension. |
== References == | == References == | ||
* Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, 2007 | * Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry], Meboo, 2007 |
Revision as of 11:39, 26 July 2008
For a list of points in Euclidean vector space, distance-square between points and is defined
Euclidean distance between points must satisfy the defining requirements imposed upon any metric space: [Dattorro, ch.5.2]
namely, for Euclidean metric in
- (nonnegativity)
- (self-distance)
- (symmetry)
- (triangle inequality)
Fifth property of the Euclidean metric (relative-angle inequality)
Augmenting the four fundamental Euclidean metric properties in , for all , , and for distinct points , the inequalities
where is the angle between vectors at vertex , must be satisfied at each point regardless of affine dimension.
References
- Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo, 2007