I'm reading Penrose's: Road To Reality. First he gives the Lambert formula and later, he says that if you want, you can include the $C$ of the Lambert area formula. But It's not clear why I would include that, from his presentation, it seems that the $C$ has to do with scaling but It's not clear what's the role of the scaling in hyperbolic geometry.
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Have you gotten to the part of the book in which Penrose explains why he "shall refer to the quantity $C^{-1/2}$ as the pseudo-radius of the geometry"? – Blue Aug 15 '15 at 20:39
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@Blue Not yet. I thought I ought to know, mainly because there is an exercise on this page. – Red Banana Aug 15 '15 at 21:11
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$C$ is related to the curvature. If the model being used is the upper half plane, the curvature is $-1$ and $C=1.$ A book that gives full detail on other choices is George Martin, The Foundations of Geometry and the Non-Euclidean Plane. The quantity that Martin calls the "distance scale" $k$ is defined in terms of a relationship among horocycles...
It may help to consider trigonometry on the sphere, then vary the radius of the sphere. Very similar, but instead of the angle sum being below $\pi,$ on the sphere it is always above $\pi.$
Will Jagy
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