Fifth Property of the Euclidean Metric
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- | 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. | + | 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. |
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== 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 21:36, 30 October 2007
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