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The angle of parallelism in [[Lobachevskii geometry|Lobachevskii geometry]] is a function that expresses the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600201.png" /> between the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600202.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600203.png" />) (see Fig.) and the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600204.png" /> perpendicular to a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600205.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600206.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600207.png" />) in terms of the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600208.png" /> of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l0600209.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002010.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002011.png" /> is a positive constant that corresponds to the scale of measurement of distances.
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== Angle of parallelism ==
 +
 
 +
The angle of parallelism in [[Lobachevskii geometry]] is a function that expresses the angle  $  \alpha $
 +
between the line  $  u _ {1} $ (or  $  u _ {2} $)
 +
(see Fig.) and the segment  $  OA $
 +
perpendicular to a line  $  a $
 +
parallel to  $  u _ {1} $(
 +
or  $  u _ {2} $)
 +
in terms of the length  $  l $
 +
of the segment  $  OA $:
 +
 
 +
$$
 +
\alpha  =  \Pi(l)  =  2  { \mathop{\rm arc}  \mathop{\rm tan} }  e ^ {- l / R } ,
 +
$$
 +
 
 +
where $R$ is a positive constant that corresponds to the scale of measurement of distances.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060020a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060020a.gif" />
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Figure: l060020a
 
Figure: l060020a
  
The Lobachevskii function is a continuous monotone decreasing function with values between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002012.png" /> and 0:
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The Lobachevskii function is a continuous monotone decreasing function with values between $\pi/2$ and 0:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002013.png" /></td> </tr></table>
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$$
 +
\lim\limits _ {l \rightarrow 0 }  \Pi ( l)  =
 +
\frac \pi {2}
 +
,\  \lim\limits _ {
 +
l \rightarrow \infty }  \Pi ( l)  = 0.
 +
$$
  
 
It was introduced by N.I. Lobachevskii in 1826.
 
It was introduced by N.I. Lobachevskii in 1826.
  
====References====
+
== Second meaning ==
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of geometry" , '''1''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,  "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR></table>
 
  
The special function (cf. [[Special functions|Special functions]]) defined for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002014.png" /> by
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The special function (cf. [[Special functions]]) defined for real $x$
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by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002015.png" /></td> </tr></table>
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$$
 +
L(x) = - \int\limits_{0}^{x} \mathop{\rm ln}  \cos t dt .
 +
$$
  
 
The Lobachevskii function can be represented as a series
 
The Lobachevskii function can be represented as a series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002016.png" /></td> </tr></table>
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$$
 +
L(x)  = x  \mathop{\rm ln}  2 - \frac{1}{2}
 +
\sum _ { k= 1} ^  {\infty }  (- 1)  ^ {k- 1} \frac{\sin  2kx }{k  ^ {2} } .
 +
$$
  
 
The main relations are:
 
The main relations are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002017.png" /></td> </tr></table>
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$$
 +
L ( - x )  = - L ( x) ,\  - \frac \pi {2} \leq  x \leq  \frac \pi {2},
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002018.png" /></td> </tr></table>
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$$
 +
L ( \pi - x )  = \pi  \mathop{\rm ln}  2 - L ( x) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060020/l06002019.png" /></td> </tr></table>
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$$
 +
L ( \pi + x )  = \pi  \mathop{\rm ln}  2 + L ( x) .
 +
$$
  
 
It was introduced by N.I. Lobachevskii in 1829.
 
It was introduced by N.I. Lobachevskii in 1829.
  
====References====
+
==Comments==
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Ryzhik,  I.S. Gradshtein,  "Tables of integrals, series, and products" , Acad. Press  (1980)  (Translated from Russian)</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
 
For the Lobachevskii function in the sense of 1) (i.e. the angle of parallelism) see also [[#References|[a1]]]–[[#References|[a4]]].
 
For the Lobachevskii function in the sense of 1) (i.e. the angle of parallelism) see also [[#References|[a1]]]–[[#References|[a4]]].
  
 
For Lobachevskii's function as defined in 2) see also [[#References|[a5]]].
 
For Lobachevskii's function as defined in 2) see also [[#References|[a5]]].
  
====References====
+
==References==
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Greenberg,  "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Parallel lines"  ''Canad. Math. Bull.'' , '''21'''  (1978)  pp. 385–397</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Bonola,  "Non-Euclidean geometry" , Dover, reprint  (1955)  (Translated from Italian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.S.M. Coxeter,  "Twelve geometric esays" , Carbondale  (1968)  pp. Chapt. 1</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of geometry" , '''1''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,  "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Ryzhik,  I.S. Gradshtein,  "Tables of integrals, series, and products" , Acad. Press  (1980)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Greenberg,  "Euclidean and non-Euclidean geometries" , Freeman  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Parallel lines"  ''Canad. Math. Bull.'' , '''21'''  (1978)  pp. 385–397</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Bonola,  "Non-Euclidean geometry" , Dover, reprint  (1955)  (Translated from Italian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.S.M. Coxeter,  "Twelve geometric esays" , Carbondale  (1968)  pp. Chapt. 1</TD></TR></table>
 +
 
 +
{{OldImage}}

Latest revision as of 12:22, 1 May 2023


Angle of parallelism

The angle of parallelism in Lobachevskii geometry is a function that expresses the angle $ \alpha $ between the line $ u _ {1} $ (or $ u _ {2} $) (see Fig.) and the segment $ OA $ perpendicular to a line $ a $ parallel to $ u _ {1} $( or $ u _ {2} $) in terms of the length $ l $ of the segment $ OA $:

$$ \alpha = \Pi(l) = 2 { \mathop{\rm arc} \mathop{\rm tan} } e ^ {- l / R } , $$

where $R$ is a positive constant that corresponds to the scale of measurement of distances.

Figure: l060020a

The Lobachevskii function is a continuous monotone decreasing function with values between $\pi/2$ and 0:

$$ \lim\limits _ {l \rightarrow 0 } \Pi ( l) = \frac \pi {2} ,\ \lim\limits _ { l \rightarrow \infty } \Pi ( l) = 0. $$

It was introduced by N.I. Lobachevskii in 1826.

Second meaning

The special function (cf. Special functions) defined for real $x$ by

$$ L(x) = - \int\limits_{0}^{x} \mathop{\rm ln} \cos t dt . $$

The Lobachevskii function can be represented as a series

$$ L(x) = x \mathop{\rm ln} 2 - \frac{1}{2} \sum _ { k= 1} ^ {\infty } (- 1) ^ {k- 1} \frac{\sin 2kx }{k ^ {2} } . $$

The main relations are:

$$ L ( - x ) = - L ( x) ,\ - \frac \pi {2} \leq x \leq \frac \pi {2}, $$

$$ L ( \pi - x ) = \pi \mathop{\rm ln} 2 - L ( x) , $$

$$ L ( \pi + x ) = \pi \mathop{\rm ln} 2 + L ( x) . $$

It was introduced by N.I. Lobachevskii in 1829.

Comments

For the Lobachevskii function in the sense of 1) (i.e. the angle of parallelism) see also [a1][a4].

For Lobachevskii's function as defined in 2) see also [a5].

References

[1] V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian)
[2] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[1] I.M. Ryzhik, I.S. Gradshtein, "Tables of integrals, series, and products" , Acad. Press (1980) (Translated from Russian)
[a1] M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)
[a2] H.S.M. Coxeter, "Parallel lines" Canad. Math. Bull. , 21 (1978) pp. 385–397
[a3] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1957)
[a4] R. Bonola, "Non-Euclidean geometry" , Dover, reprint (1955) (Translated from Italian)
[a5] H.S.M. Coxeter, "Twelve geometric esays" , Carbondale (1968) pp. Chapt. 1


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How to Cite This Entry:
Lobachevskii function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lobachevskii_function&oldid=15981
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article