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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

$$ 
\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 [[#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====
<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====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Du Val,   "Homographies, quaternions and rotations" , Clarendon Press  (1964)</TD></TR></table>