Fifth Property of the Euclidean Metric
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* <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> '''('''triangle inequality''')''' | * <math>\sqrt{d_{ij}}\,\leq\,\sqrt{d_{ik_{}}}+\sqrt{d_{kj}}~,~~i\!\neq\!j\!\neq\!k</math> '''('''triangle inequality''')''' | ||
- | where <math>\sqrt{d_{ij}}</math> is the Euclidean metric in <math>\mathbb{R}^n</math> | + | where <math>\sqrt{d_{ij}}</math> is the Euclidean metric in <math>\mathbb{R}^n</math> |
==Fifth property of the Euclidean metric == | ==Fifth property of the Euclidean metric == |
Revision as of 18:38, 17 October 2007
For a list of points in Euclidean vector space, distance-square between points and is defined
Euclidean distance must satisfy the requirements imposed by any metric space:
- (nonnegativity)
- (self-distance)
- (symmetry)
- (triangle inequality)
where is the Euclidean metric in
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.