Potential vorticity: Difference between revisions

Revision as of 00:05, 5 December 2013

Define $M_{k}^{2}$ as the 2-dimensional metric space of constant curvature $k$ . So, for example, $M_{0}^{2}$ is the Euclidean plane, $M_{1}^{2}$ is the surface of the unit sphere, and $M_{- 1}^{2}$ is the hyperbolic plane.

Let $X$ be a metric space. Let $T$ be a triangle in $X$ , with vertices $p$ , $q$ and $r$ . A comparison triangle $T *$ in $M_{k}^{2}$ for $T$ is a triangle in $M_{k}^{2}$ with vertices $p^{'}$ , $q^{'}$ and $r^{'}$ such that $d (p, q) = d (p^{'}, q^{'})$ , $d (p, r) = d (p^{'}, r^{'})$ and $d (r, q) = d (r^{'}, q^{'})$ .

Such a triangle is unique up to isometry.

The interior angle of $T *$ at $p^{'}$ is called the comparison angle between $q$ and $r$ at $p$ . This is well-defined provided $q$ and $r$ are both distinct from $p$ .

References

M Bridson & A Haefliger - Metric Spaces Of Non-Positive Curvature, ISBN 3-540-64324-9

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+Define <math>M_{k}^{2}</math> as the [[2-dimensional]] [[metric space]] of [[constant curvature]] <math>k</math>. So, for example, <math>M_{0}^{2}</math> is the [[Euclidean plane]], <math>M_{1}^{2}</math> is the surface of the [[unit sphere]], and <math>M_{-1}^{2}</math> is the [[Hyperbolic geometry|hyperbolic plane]].
+Let <math>X</math> be a [[metric space]]. Let <math>T</math> be a [[triangle]] in <math>X</math>, with vertices <math>p</math>, <math>q</math> and <math>r</math>. A '''comparison triangle''' <math>T*</math> in <math>M_{k}^{2}</math> for <math>T</math> is a triangle in <math>M_{k}^{2}</math> with vertices <math>p'</math>, <math>q'</math> and <math>r'</math> such that <math>d(p,q) = d(p',q')</math>, <math>d(p,r) = d(p',r')</math> and <math>d(r,q) = d(r',q')</math>.
+Such a triangle is unique up to [[isometry]].
+The [[interior angle]] of <math>T*</math> at <math>p'</math> is called the '''comparison angle''' between <math>q</math> and <math>r</math> at <math>p</math>. This is well-defined provided <math>q</math> and <math>r</math> are both distinct from <math>p</math>.
+==References==
+* M Bridson & [[A Haefliger]] - ''Metric Spaces Of Non-Positive [[Curvature]]'', ISBN 3-540-64324-9
+[[Category:Metric geometry]]

Potential vorticity: Difference between revisions

Revision as of 00:05, 5 December 2013

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