We know the spherical surface invariant in $\mathbb R^3$

$$ \dfrac{\sin a }{\sin A } =\dfrac {\sin b }{\sin B } =\dfrac{\sin c }{\sin C } =\dfrac{\sqrt{1-\cos^2 A -\cos^2 B-\cos^2 C +2\cos A \cos B \cos C}}{\sin A \sin B \sin C}$$

but as yet not its geometrical significance, interpretation or at least an intuition towards the meaning of this non-dimensional scalar.

Also in any model of hyperbolic surface trigonometry what is a mirroring geometrical meaning of the invariant

$$ \dfrac{\sinh a }{\sin A } =\dfrac {\sinh b }{\sin B } =\dfrac{\sinh c }{\sin C } = \dfrac{\sqrt{\cos^2 A +\cos^2 B +\cos^2 C -2\cos A \cos B \cos C-1}}{\sin A \sin B \sin C} ?$$

$ (a,b,c)$ is short for $ (a/R,b/R,c/R) $ where R is sphere radius or pseudospherical torsion radius.

May be the question involves Riemannian multi-dimensional relations, but am not sure.

Please help with a revisit to Elliptic/Hyperbolic Triangle Trig


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