What is the geometrical meaning of the common value in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$ in hyperbolic geometry? I know the meaning of this value only in Euclidean and spherical geometry.

EDIT, Will Jagy. The OP is looking for some fourth fairly natural real number that can be calculated from a triangle, that gives the same answer as the common value in the Law of Sines. The original question is at https://math.stackexchange.com/questions/69345/the-law-of-sines-in-hyperbolic-geometry

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    $\begingroup$ What is the meaning in spherical geometry? $\endgroup$ – Igor Rivin Oct 4 '11 at 20:09
  • $\begingroup$ this is the same as math.stackexchange.com/questions/69345/… where I suggested that there was no easy interpretation for the common value in the Law of Sines. Evidently in spherical geometry there is some ratio of volumes associated with the specific triangle. $\endgroup$ – Will Jagy Oct 4 '11 at 20:24
  • $\begingroup$ Yes, I am searching for some help even here, maybe there is someone here who knows the answer. Sorry for the duplicate question, I can erase it if it creates problems. $\endgroup$ – zar Oct 4 '11 at 20:30
  • $\begingroup$ Anything is possible. You should give details on your favorite interpretation in the spherical case, evidently you see this as a ratio of volumes. People need to see that. And do not call something a constant when it depends on the triangle chosen. $\endgroup$ – Will Jagy Oct 4 '11 at 20:32
  • $\begingroup$ @Will: there is an answer, and the quantity does not depend on the SIDE chosen, so it is invariant under A group of transformations. $\endgroup$ – Igor Rivin Oct 4 '11 at 20:36

There is a meaning, though whether it is geometric is up to you to decide. You can see it in: http://mathworld.wolfram.com/GeneralizedLawofSines.html which is fairly incomprehensible without http://mathworld.wolfram.com/HyperbolicPolarSine.html

To see how you might derive such a thing, see this: http://arxiv.org/abs/math/0211261

  • $\begingroup$ Thanks, Igor. If you look at the MSE original question, it was a mess, in particular it completely misled me. $\endgroup$ – Will Jagy Oct 4 '11 at 20:53
  • $\begingroup$ @Will: I am confused enough, I will take your word for it :) $\endgroup$ – Igor Rivin Oct 4 '11 at 20:58
  • $\begingroup$ Sounds good. It seems likely that the OP wants a ratio of two volumes in $\mathbf H^3,$ which seems to me a pretty good reason for expecting nothing easy. $\endgroup$ – Will Jagy Oct 4 '11 at 21:16
  • $\begingroup$ I am not sure why (s)he would, since in the Euclidean case the common value is is the circum-diameter... $\endgroup$ – Igor Rivin Oct 4 '11 at 21:30
  • $\begingroup$ Quoting: In spherical geometry it is the ratio of two volumes related to the triangle – Apotema (zar) 2 days ago $\endgroup$ – Will Jagy Oct 4 '11 at 21:48

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