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.
The law of sines in hyperbolic geometry
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3What constant? I don't see any in $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$... – 2011-10-02
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1@J.M. Probably the thing they're all equal to. For example, in the Euclidean case this is the circumdiameter (or its reciprocal). – 2011-10-02
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1Exactly! In Euclidean geometry the constant is the circudiameter. – 2011-10-02
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1@Chris: [Doesn't look to be that simple, then...](http://mathworld.wolfram.com/GeneralizedLawofSines.html) – 2011-10-02
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1In spherical geometry it is the ratio of two volumes related to the triangle – 2011-10-02
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0the 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. – 2011-10-02
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1In Euclidean geometry if you write $(\sin A_i)/a_i=k$ for $i=1,2,3$, then $k$ is the _reciprocal_ of the circumdiameter. – 2011-10-03
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6@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. – 2011-10-03
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1I 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. – 2011-10-03
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0@ Apotema. Hope people start saying, "Law of hyperbolic sines". – 2014-09-06