Quasi-elliptic space
A projective -space whose projective metric is defined by an absolute consisting of an imaginary cone (the absolute cone
) with an
-vertex (the absolute plane
) together with an imaginary
-quadric
on this
-plane (the absolute quadric
); it is denoted by the symbol
,
. A quasi-elliptic space is of more general projective type in comparison with a Euclidean space and a co-Euclidean space; the metrics of the latter are obtained from those of the former. A quasi-elliptic space is a particular case of a semi-elliptic space. For
, the absolute cone is a pair of coincident
-planes coinciding with the
-absolute plane
, while the absolute coincides with the absolute of Euclidean
-space. For
, the cone
is a cone with a point vertex and the absolute in this case is the same as that of the co-Euclidean
-space. When
, the cone
is a pair of imaginary
-planes. In particular, the cone
of the quasi-elliptic three-space
is a pair of imaginary two-planes, the line (the
-plane)
is the real line of their intersection, while the quadric
is a pair of imaginary points on
.
The distance between two points
and
is defined in case the line
does not intersect the
-plane
by the relation
![]() |
where
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are the vectors of the points and
,
is the linear operator defining the scalar product in the space of these vectors and
is a real number; in case
intersects
, the distance
between these points is defined by means of the distance between the vectors of the points
and
:
![]() |
![]() |
![]() |
where is the linear operator defining the scalar product in the space of these vectors.
The angle between two planes whose -plane of intersection does not intersect the
-plane
is defined as the (normalized) distance between the corresponding points in the dual quasi-elliptic space
, in which the coordinates are numerically equal or proportional to the projective coordinates of the planes in
. If the
-plane of intersection of two given planes intersects the
-plane
, then the angle between the planes is in this case again defined by the numerical distance. When
the angles between the planes are the angles between the lines.
The motions of the quasi-elliptic space are the collineations of this space that take the cone
into the plane
and the quadric
into itself. The group of motions is a Lie group and the motions are described by orthogonal operators. In the quasi-elliptic space
, which is self-dual, co-motions are defined as the correlations that take each pair of points into two
-planes the angle between which is proportional to the distance between the points, and each pair of
-planes into two points the distance between which is proportional to the angle between the planes. The motions and co-motions of
form a group, which is a Lie group. The geometry of the
-plane
is Euclidean geometry, while the geometry of the
-plane
is the same as that of the co-Euclidean plane.
The geometry of the -space
is defined by an elliptic projective metric on lines that is co-Euclidean on planes and Euclidean in bundles of planes. The geometry of the
-space
is Euclidean, while the geometry of the
-space
is the same as that of the co-Euclidean
-space. The space
with radius of curvature
is isometric to the connected group of motions of the Euclidean
-space with a specially introduced metric. The connected group of motions of the quasi-Euclidean space
is isomorphic to the direct product of two connected groups of motions of the Euclidean
-plane.
References
[1] | B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian) |
Comments
References
[a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
[a2] | O. Giering, "Vorlesungen über höhere Geometrie" , Vieweg (1982) |
Quasi-elliptic space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-elliptic_space&oldid=16290