Fifth Property of the Euclidean Metric
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(Difference between revisions)
Line 4: | Line 4: | ||
<math>\begin{array}{rl}d_{ij} | <math>\begin{array}{rl}d_{ij} | ||
\!\!&=\,\|x_i-_{}x_j\|^2 | \!\!&=\,\|x_i-_{}x_j\|^2 | ||
- | ~=~(x_i-_{}x_j)^T(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^ | + | ~=~(x_i-_{}x_j)^{\rm T}(x_i-_{}x_j)~=~\|x_i\|^2+\|x_j\|^2-2_{}x^{\rm T}_i\!x_j\\\\ |
- | &=\,\left[x_i^T\quad x_j^T\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right] | + | &=\,\left[x_i^{\rm T}\quad x_j^{\rm T}\right]\left[\begin{array}{rr}\!I&-I\\\!-I&I\end{array}\right] |
\left[\!\!\begin{array}{cc}x_i\\x_j\end{array}\!\!\right] | \left[\!\!\begin{array}{cc}x_i\\x_j\end{array}\!\!\right] | ||
\end{array}</math> | \end{array}</math> |
Revision as of 21:35, 4 December 2011
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, 2005