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Special coordinates in the rotation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210601.png" /> of three-dimensional space, the construction of which is based in the final analysis on the relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210602.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210603.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210604.png" /> unitary matrices with determinant 1. There exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210605.png" /> which is an [[Epimorphism|epimorphism]] by virtue of its algebraic properties and a double [[Covering|covering]] by virtue of its topological properties. (Restricted to some neighbourhood of the identity matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210606.png" /> is an isomorphism; in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210608.png" /> are locally isomorphic.) Each matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c0210609.png" /> may be written as
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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/c/c021/c021060/c02106010.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106011.png" /> are complex numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106012.png" />. These are taken to be the Cayley–Klein parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106013.png" />. (The term is sometimes used for all four elements of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106014.png" />.) As the actual construction of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106015.png" /> with the above properties may be accomplished in various ways, different authors define the Cayley–Klein parameters in slightly different ways (see [[#References|[2]]], [[#References|[3]]]).
+
Special coordinates in the rotation group  $  \mathop{\rm SO} ( 3) $
 +
of three-dimensional space, the construction of which is based in the final analysis on the relationship between  $  \mathop{\rm SO} ( 3) $
 +
and the group  $  \mathop{\rm SU} ( 2) $
 +
of $  2 \times 2 $
 +
unitary matrices with determinant 1. There exists a mapping $  \phi :   \mathop{\rm SU} ( 2) \rightarrow  \mathop{\rm SO} ( 3) $
 +
which is an [[Epimorphism|epimorphism]] by virtue of its algebraic properties and a double [[Covering|covering]] by virtue of its topological properties. (Restricted to some neighbourhood of the identity matrix,  $  \phi $
 +
is an isomorphism; in other words,  $  \mathop{\rm SO} ( 3) $
 +
and  $  \mathop{\rm SU} ( 2) $
 +
are locally isomorphic.) Each matrix  $  V \in  \mathop{\rm SU} ( 2) $
 +
may be written as
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106016.png" /> is not a true isomorphism, but only a double covering, it is impossible to define the Cayley–Klein parameters as (continuous) coordinates on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106017.png" />; this can be done only locally. However, the Cayley–Klein parameters may nevertheless be used to study processes of rotation in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106018.png" /> is a continuous function of a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106019.png" /> (and there is no necessity to restrict the domain of possible values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106020.png" /> in any way). Indeed, if some fixed value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106021.png" /> has been chosen at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106022.png" />, the corresponding values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106023.png" /> are uniquely defined by continuity for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106024.png" />. (The fact that the complete inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106025.png" /> is double-valued intrudes only in the observation that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106026.png" /> not only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106027.png" /> but also when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106028.png" />.) Thus the Cayley–Klein parameters can be applied to investigate the motion of a rigid body with a fixed point (the configuration space of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106029.png" />). This approach was adopted in [[#References|[1]]], but did not achieve popularity.
+
$$
 +
\left \|
 +
\begin{array}{rr}
 +
\alpha  &\beta  \\
 +
- \overline \beta \; &\overline \alpha \;  \\
 +
\end{array}
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106030.png" /> is isomorphic to the group of quaternions with norm 1 (cf. [[Quaternion|Quaternion]]); hence, by going over from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106031.png" /> to the corresponding quaternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106032.png" />, one can replace the Cayley–Klein parameters by the Euler–Rodriguez parameters — four real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021060/c02106034.png" />. These stand in a simple relationship to the Cayley–Klein parameters (see [[#References|[1]]], [[#References|[3]]]) and possess the same  "double-valuedness"  property (for the history of the problem see [[#References|[1]]]). It was essentially in this context that attention was first directed to two-valued representations of the rotation group (see [[Spinor|Spinor]]).
+
\right \| ,
 +
$$
 +
 
 +
where  $  \alpha , \beta $
 +
are complex numbers such that  $  | \alpha |  ^ {2} + | \beta |  ^ {2} = 1 $.
 +
These are taken to be the Cayley–Klein parameters of  $  A = \phi ( V) $.
 +
(The term is sometimes used for all four elements of the matrix  $  V $.)
 +
As the actual construction of a mapping  $  \phi $
 +
with the above properties may be accomplished in various ways, different authors define the Cayley–Klein parameters in slightly different ways (see [[#References|[2]]], [[#References|[3]]]).
 +
 
 +
Since  $  \phi $
 +
is not a true isomorphism, but only a double covering, it is impossible to define the Cayley–Klein parameters as (continuous) coordinates on all of  $  \mathop{\rm SO} ( 3) $;
 +
this can be done only locally. However, the Cayley–Klein parameters may nevertheless be used to study processes of rotation in which  $  A $
 +
is a continuous function of a real parameter  $  t $(
 +
and there is no necessity to restrict the domain of possible values of  $  A $
 +
in any way). Indeed, if some fixed value  $  V ( t _ {0} ) \in \phi  ^ {-} 1 ( A ( t _ {0} )) $
 +
has been chosen at  $  t = t _ {0} $,
 +
the corresponding values of  $  V ( t) $
 +
are uniquely defined by continuity for all  $  t $.
 +
(The fact that the complete inverse  $  \phi  ^ {-} 1 $
 +
is double-valued intrudes only in the observation that  $  A ( t) = A ( s) $
 +
not only when  $  V ( t) = V ( s) $
 +
but also when  $  V ( t) = - V ( s) $.)
 +
Thus the Cayley–Klein parameters can be applied to investigate the motion of a rigid body with a fixed point (the configuration space of which is  $  \mathop{\rm SO} ( 3) $).  
 +
This approach was adopted in [[#References|[1]]], but did not achieve popularity.
 +
 
 +
The group  $  \mathop{\rm SU} ( 2) $
 +
is isomorphic to the group of quaternions with norm 1 (cf. [[Quaternion|Quaternion]]); hence, by going over from $  V $
 +
to the corresponding quaternion $  \rho + \lambda i + \mu j + \nu k $,  
 +
one can replace the Cayley–Klein parameters by the Euler–Rodriguez parameters — four real numbers $  \rho , \lambda , \mu , \nu $
 +
such that $  \rho  ^ {2} + \lambda  ^ {2} + \mu  ^ {2} + \nu  ^ {2} = 1 $.  
 +
These stand in a simple relationship to the Cayley–Klein parameters (see [[#References|[1]]], [[#References|[3]]]) and possess the same  "double-valuedness"  property (for the history of the problem see [[#References|[1]]]). It was essentially in this context that attention was first directed to two-valued representations of the rotation group (see [[Spinor|Spinor]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  A. Sommerfeld,  "Ueber die Theorie des Kreises" , '''1–2''' , Teubner  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Synge,  "Classical dynamics" , ''Handbuch der Physik'' , '''3/1''' , Springer  (1960)  pp. 1–225</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Klein,  A. Sommerfeld,  "Ueber die Theorie des Kreises" , '''1–2''' , Teubner  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Goldstein,  "Classical mechanics" , Addison-Wesley  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Synge,  "Classical dynamics" , ''Handbuch der Physik'' , '''3/1''' , Springer  (1960)  pp. 1–225</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Du Val,  "Homographies, quaternions and rotations" , Clarendon Press  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Du Val,  "Homographies, quaternions and rotations" , Clarendon Press  (1964)</TD></TR></table>

Revision as of 16:43, 4 June 2020


Special coordinates in the rotation group $ \mathop{\rm SO} ( 3) $ of three-dimensional space, the construction of which is based in the final analysis on the relationship between $ \mathop{\rm SO} ( 3) $ and the group $ \mathop{\rm SU} ( 2) $ of $ 2 \times 2 $ unitary matrices with determinant 1. There exists a mapping $ \phi : \mathop{\rm SU} ( 2) \rightarrow \mathop{\rm SO} ( 3) $ which is an epimorphism by virtue of its algebraic properties and a double covering by virtue of its topological properties. (Restricted to some neighbourhood of the identity matrix, $ \phi $ is an isomorphism; in other words, $ \mathop{\rm SO} ( 3) $ and $ \mathop{\rm SU} ( 2) $ are locally isomorphic.) Each matrix $ V \in \mathop{\rm SU} ( 2) $ may be written as

$$ \left \| \begin{array}{rr} \alpha &\beta \\ - \overline \beta \; &\overline \alpha \; \\ \end{array} \right \| , $$

where $ \alpha , \beta $ are complex numbers such that $ | \alpha | ^ {2} + | \beta | ^ {2} = 1 $. These are taken to be the Cayley–Klein parameters of $ A = \phi ( V) $. (The term is sometimes used for all four elements of the matrix $ V $.) As the actual construction of a mapping $ \phi $ with the above properties may be accomplished in various ways, different authors define the Cayley–Klein parameters in slightly different ways (see [2], [3]).

Since $ \phi $ is not a true isomorphism, but only a double covering, it is impossible to define the Cayley–Klein parameters as (continuous) coordinates on all of $ \mathop{\rm SO} ( 3) $; this can be done only locally. However, the Cayley–Klein parameters may nevertheless be used to study processes of rotation in which $ A $ is a continuous function of a real parameter $ t $( and there is no necessity to restrict the domain of possible values of $ A $ in any way). Indeed, if some fixed value $ V ( t _ {0} ) \in \phi ^ {-} 1 ( A ( t _ {0} )) $ has been chosen at $ t = t _ {0} $, the corresponding values of $ V ( t) $ are uniquely defined by continuity for all $ t $. (The fact that the complete inverse $ \phi ^ {-} 1 $ is double-valued intrudes only in the observation that $ A ( t) = A ( s) $ not only when $ V ( t) = V ( s) $ but also when $ V ( t) = - V ( s) $.) Thus the Cayley–Klein parameters can be applied to investigate the motion of a rigid body with a fixed point (the configuration space of which is $ \mathop{\rm SO} ( 3) $). This approach was adopted in [1], but did not achieve popularity.

The group $ \mathop{\rm SU} ( 2) $ is isomorphic to the group of quaternions with norm 1 (cf. Quaternion); hence, by going over from $ V $ to the corresponding quaternion $ \rho + \lambda i + \mu j + \nu k $, one can replace the Cayley–Klein parameters by the Euler–Rodriguez parameters — four real numbers $ \rho , \lambda , \mu , \nu $ such that $ \rho ^ {2} + \lambda ^ {2} + \mu ^ {2} + \nu ^ {2} = 1 $. These stand in a simple relationship to the Cayley–Klein parameters (see [1], [3]) and possess the same "double-valuedness" property (for the history of the problem see [1]). It was essentially in this context that attention was first directed to two-valued representations of the rotation group (see Spinor).

References

[1] F. Klein, A. Sommerfeld, "Ueber die Theorie des Kreises" , 1–2 , Teubner (1965)
[2] H. Goldstein, "Classical mechanics" , Addison-Wesley (1953)
[3] J.L. Synge, "Classical dynamics" , Handbuch der Physik , 3/1 , Springer (1960) pp. 1–225

Comments

References

[a1] P. Du Val, "Homographies, quaternions and rotations" , Clarendon Press (1964)
How to Cite This Entry:
Cayley-Klein parameters. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Klein_parameters&oldid=46287
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article