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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.

J. M. ain't a mathematician
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Apotema
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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. ain't a mathematician Oct 02 '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 02 '11 at 21:20
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    Exactly! In Euclidean geometry the constant is the circudiameter. – Apotema Oct 02 '11 at 21:22
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    @Chris: Doesn't look to be that simple, then... – J. M. ain't a mathematician Oct 02 '11 at 21:24
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    In spherical geometry it is the ratio of two volumes related to the triangle – Apotema Oct 02 '11 at 21:27
  • the equation above is incorrect, just erase the $=k$ part and then it is alright. The three ratios are indeed equal, but not set equal to any fourth item. – Will Jagy Oct 02 '11 at 23:23
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    In Euclidean geometry if you write $(\sin A_i)/a_i=k$ for $i=1,2,3$, then $k$ is the reciprocal of the circumdiameter. – Michael Hardy Oct 03 '11 at 00:35
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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 03 '11 at 00:37
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    I see. The phrasing threw me, the use of the word "constant" and of the letter "k" suggested something specific. It is unlikely that there is anything else understandable related to a triangle that also gives the common value in the law of Sines. – Will Jagy Oct 03 '11 at 01:51
  • @ Apotema. Hope people start saying, "Law of hyperbolic sines". – Narasimham Sep 06 '14 at 11:10
  • @J.M.ain'tamathematician, thank you for the link that you post. Given the gsin() defined in that article, what's the geometric meaning of the expression ( gsin(S_0) gsin(S_1) gsin(S_2) ) ? I post this new question here: https://math.stackexchange.com/questions/4986401/whats-the-geometric-meaning-of-the-common-value-in-the-spherical-law-of-sines – yo-yos Oct 18 '24 at 15:42

2 Answers2

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"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.

Will Jagy
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As given by Will Jagy, k must be inside the argument: $$ \frac{\sin A}{\sinh(a/k)} = \frac{\sin B}{\sinh(b/k)} = \frac{\sin C}{\sinh(c/k)} $$

This is the Law of Hyperbolic trigonometry where k is the pseudoradius, constant Gauss curvature $K= -1/k^2$. Please also refer to " Pan-geometry", a set of relations mirrored from spherical to hyerbolic, typified by (sin,cos) -> (sinh,cosh).. in Roberto Bonola's book on Non-euclidean Geometry.

There is nothing imaginary about pseudoradius.It is as real,palpable and solid as the radius of sphere in spherical trigonometry, after hyperbolic geometry has been so firmly established.

I wish practice of using $ K=-1$ should be done away with,always using $ K = -1/k^2 $ or $ K = -1/a^2 $instead.

Narasimham
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