Fifth Property of the Euclidean Metric
From Wikimization
(Difference between revisions)
Line 8: | Line 8: | ||
\end{array}</math> | \end{array}</math> | ||
- | [[Euclidean distance]] must satisfy the requirements imposed by any metric space | + | [[Euclidean distance]] must satisfy the requirements imposed by any metric space. |
{{harvtxt|Dattorro|2007, ch.5.2}} | {{harvtxt|Dattorro|2007, ch.5.2}} |
Revision as of 20:15, 23 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.