Potential vorticity

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 $p^{\prime }$, $q^{\prime }$ and $r^{\prime }$ such that $d(p,q)=d(p^{\prime },q^{\prime })$, $d(p,r)=d(p^{\prime },r^{\prime })$ and $d(r,q)=d(r^{\prime },q^{\prime })$.

Such a triangle is unique up to isometry.

The interior angle of $T*$ at $p^{\prime }$ 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