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.
Asked
Active
Viewed 1,123 times
5
-
3What 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
-
2@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
-
1Exactly! In Euclidean geometry the constant is the circudiameter. – Apotema Oct 02 '11 at 21:22
-
1@Chris: Doesn't look to be that simple, then... – J. M. ain't a mathematician Oct 02 '11 at 21:24
-
1In 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
-
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. – Michael Hardy Oct 03 '11 at 00:35
-
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. – Michael Hardy Oct 03 '11 at 00:37
-
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. – Will Jagy Oct 03 '11 at 01:51
-
@ Apotema. Hope people start saying, "Law of hyperbolic sines". – Narasimham Sep 06 '14 at 11:10
2 Answers
5
"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
- 139,541
0
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
- 40,495