Hyperbolic law of cosines

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]

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”).

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:

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

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:

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

(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]

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

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]

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

• Hyperbolic law of sines
• Hyperbolic triangle trigonometry
• History of Lorentz transformations

• Non Euclidean Geometry, Math Wiki at TU Berlin