Fifth Property of the Euclidean Metric
From Wikimization
(Difference between revisions)
Line 1: | Line 1: | ||
- | 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} |
Revision as of 01:01, 25 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.