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

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What constant? I don't see any in $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$... –  J. M. Oct 2 '11 at 21:15
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@J.M. Probably the thing they're all equal to. For example, in the Euclidean case this is the circumdiameter (or its reciprocal). –  Chris Eagle Oct 2 '11 at 21:20
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Exactly! In Euclidean geometry the constant is the circudiameter. –  Apotema Oct 2 '11 at 21:22
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@Chris: Doesn't look to be that simple, then... –  J. M. Oct 2 '11 at 21:24
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@WillJagy: You seem to miss the point. The common value is the definition of $k$. The question is: what is the geometric meaning of $k$? In Euclidean geometry, it's well known. –  Michael Hardy Oct 3 '11 at 0:37

1 Answer 1

"k" is the "distance scale," traditionally taken to be 1, therefore in a space of curvature $-1,$ as the curvature is $-1/k^2.$ In this and other ways, $k$ appears as a sort of imaginary radius. Note the curvature of the ordinary sphere of radius $r$ is $1/r^2.$

Oh, $k$ does NOT appear in the place you indicate in the Law of Sines. Erase it!

If you want to allow other $k,$ the correct Law is $$ \frac{\sin A}{\sinh(a/k)} = \frac{\sin B}{\sinh(b/k)} = \frac{\sin C}{\sinh(c/k)} $$

The actual meaning of $k$ is a relation between curves called horocycles. But, for something easier, the area of a geodesic triangle is its angular defect multiplied by $k^2.$

The easiest introduction I know to these matters is MY_ARTICLE

EDIT: evidently Apotema wanted some other geometric number associated with a triangle that gives the same number as the common value in the Law of Sines. I cannot imagine anything understandable that does that. See the article by Milnor on the first 150 years of hyperbolic geometry: MILNOR. There is no nice expression for the volume of a tetrahedron in $\mathbf H^3.$ I've got to think about whether I even know the volume of a geodesic sphere in $\mathbf H^3.$ Had to look it up, $$ V = 2 \, \pi \, ( \, \sinh r \; \cosh r \; \; - \; \; r ) = \pi \sinh(2r) - 2 \pi r, $$ and that the Taylor series of this around $r=0$ has first term $\frac{4}{3} \pi r^3,$ as is required in the small.

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