Английская Википедия:Hyperbolic law of cosines

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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. It can also be related to the relativistic velocity addition formula.

History

Describing relations of hyperbolic geometry, Franz Taurinus showed in 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:

<math display="block">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)}</math>

which was also shown by Nikolai Lobachevsky (1830):

<math display="block">\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] </math>

Ferdinand Minding gave it in relation to surfaces of constant negative curvature:

<math display="block">\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</math>

as did Delfino Codazzi in 1857:

<math display="block">\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]</math>

The relation to relativity using rapidity was shown by Arnold Sommerfeld in 1909 and Vladimir Varićak in 1910.

Hyperbolic laws of cosines

Take a hyperbolic plane whose Gaussian curvature is <math display="inline">-\frac{1}{k^{2}}</math>. Given a hyperbolic triangle <math>ABC</math> with angles <math>\alpha,\beta,\gamma</math> and side lengths <math>BC = a</math>, <math>AC = b</math>, and <math>AB = c</math>, the following two rules hold. The first is an analogue of Euclidean law of cosines, expressing the length of one side in terms of the other two and the angle between the latter:

The second law has no Euclidean analogue, since it expresses the fact that lengths of sides of a hyperbolic triangle are determined by the interior angles:

<math display="block">\cos\alpha=-\cos\beta\cos\gamma+\sin\beta\sin\gamma\cosh\frac{a}{k}.</math>

Houzel indicates that the hyperbolic law of cosines implies the angle of parallelism in the case of an ideal hyperbolic triangle:

Hyperbolic law of Haversines

In cases where <math>a/k</math> 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:

<math display="block">\sinh^{2}\frac{a}{2k}=\sinh^{2}\frac{b-c}{2k}+\sinh\frac{b}{k}\sinh\frac{c}{k}\sin^{2}\frac{\alpha}{2},</math>

Relativistic velocity addition via hyperbolic law of cosines

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

Шаблон:NumBlk &= \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{align}</math>

| Шаблон:EquationRef

}}

In comparison, the velocity addition formulas of special relativity for the x and y-directions as well as under an arbitrary angle <math>\alpha</math>, where Шаблон:Mvar is the relative velocity between two inertial frames, Шаблон:Mvar the velocity of another object or frame, and Шаблон:Mvar the speed of light, is given by

<math display="block">\begin{align}

 &&
 \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{align}</math>

It turns out that this result corresponds to the hyperbolic law of cosines - by identifying <math>\left[\xi,\ \eta,\ \zeta\right]</math> with relativistic rapidities <math>{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\xi,\ \tanh\eta,\ \tanh\zeta\right]\right)} ,</math> the equations in (Шаблон:EquationNote) assume the form:

<math display="block">\begin{align}

 &&
 \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{align}</math>

See also

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

References

Bibliography

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite book
• Шаблон:Cite journal

External links

• Non Euclidean Geometry, Math Wiki at TU Berlin
• Velocity Compositions and Rapidity, at MathPages