Hyperbolic law of cosines

In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry.[1][2][3] It can also be related to the relativisic velocity addition formula.[4][5][6]

History

Describing relations of hyperbolic geometry,[7][8][9][10] it was shown by Franz Taurinus (1826) that the spherical law of cosines can be related to spheres of imaginary radius, thus he arrived at the hyperbolic law of cosines in the form:[11]

${\displaystyle A=\operatorname {arccos} {\frac {\cos \left(\alpha {\sqrt {-1}}\right)-\cos \left(\beta {\sqrt {-1}}\right)\cos \left(\gamma {\sqrt {-1}}\right)}{\sin \left(\beta {\sqrt {-1}}\right)\sin \left(\gamma {\sqrt {-1}}\right)}}}$

which was also shown by Nikolai Lobachevsky (1830):[12]

${\displaystyle \cos A\sin b\sin c-\cos b\cos c=\cos a;\quad [a,\ b,\ c]\rightarrow \left[a{\sqrt {-1}},\ b{\sqrt {-1}},\ c{\sqrt {-1}}\right]}$

Ferdinand Minding (1840) gave it in relation to surfaces of constant negative curvature:[13]

${\displaystyle \cos a{\sqrt {k}}=\cos b{\sqrt {k}}\cdot \cos c{\sqrt {k}}+\sin b{\sqrt {k}}\cdot \sin c{\sqrt {k}}\cdot \cos A}$

as did Delfino Codazzi (1857):[14]

${\displaystyle \cos \beta \,p\left({\frac {a}{r}}\right)p\left({\frac {s}{r}}\right)=q\left({\frac {a}{r}}\right)q\left({\frac {s}{r}}\right)-q\left({\frac {\lambda }{r}}\right),\quad \left[{\frac {e^{t}-e^{-t}}{2}}=p(t),\ {\frac {e^{t}+e^{-t}}{2}}=q(t)\right]}$

The relation to relativity using rapidity was shown by Arnold Sommerfeld (1909)[15] and Vladimir Varićak (1910).[16]

Hyperbolic law of cosines

Take a hyperbolic plane whose Gaussian curvature is ${\displaystyle -{\frac {1}{k^{2}}}}$. Then given a hyperbolic triangle ${\displaystyle ABC}$ with angles ${\displaystyle \alpha ,\beta ,\gamma }$ and side lengths ${\displaystyle BC=a}$, ${\displaystyle AC=b}$, and ${\displaystyle AB=c}$, the following two rules hold:

${\displaystyle \cosh {\frac {a}{k}}=\cosh {\frac {b}{k}}\cosh {\frac {c}{k}}-\sinh {\frac {b}{k}}\sinh {\frac {c}{k}}\cos \alpha ,}$

(1)

considering the sides, while

${\displaystyle \cos \alpha =-\cos \beta \cos \gamma +\sin \beta \sin \gamma \cosh {\frac {a}{k}},}$

for the angles.

Christian Houzel (page 8) indicates that the hyperbolic law of cosines implies the angle of parallelism in the case of an ideal hyperbolic triangle:[17]

When ${\displaystyle \alpha =0}$, that is when the vertex ”A” is rejected to infinity and the sides ”BA” and ”CA” are ”parallel”, the first member equals 1; let us suppose in addition that ${\displaystyle \gamma =\pi /2}$ so that ${\displaystyle \cos \gamma =0}$ and ${\displaystyle \sin \gamma =1}$. The angle at ”B” takes a value β given by ${\displaystyle 1=\sin \beta \cosh(a/k)}$; this angle was later called ”angle of parallelism” and Lobachevsky noted it by ”F(a)” or Π(”a”).

Hyperbolic law of Haversines

In cases where ”a/k” is small, and being solved for, the numerical precision of the standard form of the hyperbolic law of cosines will drop due to rounding errors, for exactly the same reason it does in the Spherical law of cosines. The hyperbolic version of the law of haversines can prove useful in this case:

${\displaystyle \sinh ^{2}{\frac {a}{2k}}=\sinh ^{2}{\frac {b-c}{2k}}+\sinh {\frac {b}{k}}\sinh {\frac {c}{k}}\sin ^{2}{\frac {\alpha }{2}},}$

Relativistic velocity addition via hyperbolic law of cosines

Setting ${\displaystyle \left[{\tfrac {a}{k}},\ {\tfrac {b}{k}},\ {\tfrac {c}{k}}\right]=\left[\xi ,\ \eta ,\ \zeta \right]}$ in (1), and by using hyperbolic identities in terms of the hyperbolic tangent, the hyperbolic law of cosines can be written:

{\displaystyle {\begin{matrix}\cosh \xi =\cosh \eta \cosh \zeta -\sinh \eta \sinh \zeta \cos \alpha \\\downarrow \\{\frac {1}{\sqrt {1-\tanh ^{2}\xi }}}={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}{\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}-{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}{\frac {\tanh \zeta }{\sqrt {1-\tanh ^{2}\zeta }}}\cos \alpha \\\downarrow \\{\begin{aligned}\tanh \xi &={\frac {\sqrt {-\tanh ^{2}\zeta -\tanh ^{2}\eta +2\tanh \eta \tanh \zeta \cos \alpha +\left(\tanh \eta \tanh \zeta \sin \alpha \right)^{2}}}{1-\tanh \eta \tanh \zeta \cos \alpha }}\end{aligned}}\end{matrix}}}

(2)

In comparison, the velocity addition formulas of special relativity for the x and y-directions as well as under an arbitrary angle ${\displaystyle \alpha }$, where v is the relative velocity between two inertial frames, u the velocity of another object or frame, and c the speed of light, is given by[4][18]

{\displaystyle {\begin{matrix}\left[U_{x},\ U_{y}\right]=\left[{\frac {u_{x}-v}{1-{\frac {v}{c^{2}}}u_{x}}},\ {\frac {u_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c^{2}}}u_{x}}}\right]\\\hline U^{2}=U_{x}^{2}+U_{y}^{2},\ u^{2}=u_{x}^{2}+u_{y}^{2},\ \tan \alpha ={\frac {u_{y}}{u_{x}}}\\\downarrow \\{\begin{aligned}U={\frac {\sqrt {-u^{2}-v^{2}+2vu\cos \alpha +\left({\frac {vu\sin \alpha }{c}}\right){}^{2}}}{1-{\frac {v}{c^{2}}}u\cos \alpha }}\end{aligned}}\end{matrix}}}

It turns out that this result corresponds to the hyperbolic law of cosines - by identifying ${\displaystyle \left[\xi ,\ \eta ,\ \zeta \right]}$ with relativistic rapidities ${\displaystyle {\scriptstyle \left(\left[{\frac {U}{c}},\ {\frac {v}{c}},\ {\frac {u}{c}}\right]=\left[\tanh \xi ,\ \tanh \eta ,\ \tanh \zeta \right]\right)}}$, the equations in (2) assume the form:[16][5][6]

{\displaystyle {\begin{matrix}\cosh \xi =\cosh \eta \cosh \zeta -\sinh \eta \sinh \zeta \cos \alpha \\\downarrow \\{\frac {1}{\sqrt {1-{\frac {U^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {1}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}-{\frac {v/c}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\frac {u/c}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\cos \alpha \\\downarrow \\{\begin{aligned}U={\frac {\sqrt {-u^{2}-v^{2}+2vu\cos \alpha +\left({\frac {vu\sin \alpha }{c}}\right){}^{2}}}{1-{\frac {v}{c^{2}}}u\cos \alpha }}\end{aligned}}\end{matrix}}}

References

1. ^ Anderson, James W. (2005). Hyperbolic geometry (2nd ed.). London: Springer. ISBN 1-85233-934-9.
2. ^ Miles Reid & Balázs Szendröi (2005) ”Geometry and Topology”, §3.10 Hyperbolic triangles and trig, Cambridge University Press, ISBN 0-521-61325-6, MR2194744.
3. ^ Reiman, István (1999). Geometria és határterületei. Szalay Könyvkiadó és Kereskedőház Kft. ISBN 978-963-237-012-5.
4. ^ a b Pauli, Wolfgang (1921), "Die Relativitätstheorie", Encyclopädie der mathematischen Wissenschaften, 5 (2): 539–776
In English: Pauli, W. (1981) [1921]. Theory of Relativity. Fundamental Theories of Physics. 165. Dover Publications. ISBN 0-486-64152-X.
5. ^ a b Barrett, J.F. (2006), The hyperbolic theory of relativity arXiv:1102.0462
6. ^ a b Mathpages: Velocity Compositions and Rapidity
7. ^ Bonola, R. (1912). Non-Euclidean geometry: A critical and historical study of its development. Chicago: Open Court.
8. ^ Bonola (1912), p. 79 for Taurinus; p. 89 for Lobachevsky; p. 137 for Minding
9. ^ Gray, J. (1979). "Non-euclidean geometry—A re-interpretation". Historia Mathematica. 6 (3): 236–258. doi:10.1016/0315-0860(79)90124-1.
10. ^ Gray (1979), p. 242 for Taurinus; p. 244 for Lobachevsky; p. 246 for Minding
11. ^ Taurinus, Franz Adolph (1826). Geometriae prima elementa. Recensuit et novas observationes adjecit. Köln: Bachem. p. 66.
12. ^ Lobachevsky, N. (1898) [1830]. "Ueber die Anfangsgründe der Geometrie". In Engel, F.; Stäckel, P. Zwei geometrische Abhandlungen. Leipzig: Teubner. pp. 21–65.
13. ^ Minding, F. (1840). "Beiträge zur Theorie der kürzesten Linien auf krummen Flächen". Journal für die reine und angewandte Mathematik. 20: 324.
14. ^ Codazzi, D. (1857). "Intorno alle superficie le quali hanno costante il prodotto de due raggi di curvatura". Ann. Sci. Mat. Fis. 8: 351–354.
15. ^ Sommerfeld, A. (1909), "Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie" [Wikisource translation: On the Composition of Velocities in the Theory of Relativity], Verh. Der DPG, 21: 577–582
16. ^ a b Varičak, Vladimir (1912), "Über die nichteuklidische Interpretation der Relativtheorie" [On the Non-Euclidean Interpretation of the Theory of Relativity], Jahresbericht der Deutschen Mathematiker-Vereinigung, 21: 103–127
17. ^ Houzel, Christian (1992) "The Birth of Non-Euclidean Geometry", pages 3 to 21 in ”1830–1930: A Century of Geometry”, Lecture Notes in Physics #402, Springer-Verlag ISBN 3-540-55408-4 .
18. ^ Pauli (1921), p. 561