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A model of the manifold of lines in a three-dimensional elliptic space  $  S _ {3} $
on a pair of two-dimensional elliptic planes  $  S _ {2} $. 
Pairs of mutually-polar lines in  $  S _ {3} $
are represented in a one-to-one manner by pairs of diametrically-opposite points of two spheres of unit radius in the Euclidean space  $  \mathbf R  ^ {3} $. 
When one identifies diametrically-opposite points, one obtains a one-to-one representation of pairs of polar lines in  $  S _ {3} $
by points of two elliptic planes  $  S _ {2} $. 
The manifold of pairs of polar lines is homeomorphic to the topological product of these two planes  $  S _ {2} $. 
A motion of  $  S _ {3} $
is represented in the Fubini model by independent motions of the two planes  $  S _ {2} $: 
Every connected group of motions of  $  S _ {3} $
is isomorphic to the direct product of two groups of motions of  $  S _ {2} $; 
the group of motions of  $  S _ {3} $
is isomorphic to the direct product of the two groups of motions of the pair of planes  $  S _ {2} $.

A Fubini model can also be constructed for a three-dimensional hyperbolic space  $  {}  ^ {2} S _ {3} $. 
In this case one uses the [[Plücker interpretation|Plücker interpretation]] of the projective space  $  P _ {3} $
in  $  {}  ^ {3} S _ {5} $. 
The group of motions of  $  {}  ^ {2} S _ {3} $
is isomorphic to the direct product of the two groups of motions of  $  {}  ^ {1} S _ {2} $; 
it is represented in the Plücker model by the subgroup of the group of motions of  $  {}  ^ {3} S _ {5} $
consisting of the motions that map two mutually-polar hyperbolic  $  2 $-
planes into themselves. The lines of intersection of these planes with the absolute of  $  {}  ^ {3} S _ {5} $
form a family of straight line generators of a ruled quadric. The manifold of pairs of polar elliptic lines in  $  {}  ^ {2} S _ {3} $
is homeomorphic to the topological product of proper domains of the two planes  $  {}  ^ {1} S _ {2} $, 
that is, to the topological product of two discs, and the manifold of pairs of polar hyperbolic lines is homeomorphic to the product of ideal domains of the two planes  $  {}  ^ {1} S _ {2} $, 
that is, the topological product of two Möbius strips.

The model was proposed by G. Fubini [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Fubini,   ''Ann. Scuola Norm. Sup. Pisa'' , '''9'''  (1904)  pp. 1–74</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>

====Comments====
The constructions of Fubini follow easily from the decomposition of the orthogonal group  $   \mathop{\rm SO} ( 4) $
given by the quaternions: see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''II''' , Springer  (1987)  pp. Sect. 8.9</TD></TR></table>