In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension . Simple examples in a real projective space are hyperspheres (
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
s). The essential geometric properties of an ovoid
are:
# Any line intersects
in at most 2 points,
# The tangents at a point cover a hyperplane (and nothing more), and
#
contains no lines.
Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...).
An ovoid is the spatial analog of an
oval
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...
in a projective plane.
An ovoid is a special type of a ''
quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
.''
Ovoids play an essential role in constructing examples of
Möbius plane In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane.
A s ...
s and higher dimensional Möbius geometries.
Definition of an ovoid
* In a projective space of dimension a set
of points is called an ovoid, if
: (1) Any line meets
in at most 2 points.
In the case of
, the line is called a ''passing'' (or ''exterior'') ''line'', if
the line is a ''tangent line'', and if
the line is a ''secant line''.
: (2) At any point
the tangent lines through cover a hyperplane, the ''tangent hyperplane'', (i.e., a projective subspace of dimension ).
: (3)
contains no lines.
From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
*For an ovoid
and a hyperplane
, which contains at least two points of
, the subset
is an ovoid (or an oval, if ) within the hyperplane
.
For ''finite'' projective spaces of dimension (i.e., the point set is finite, the space is pappian), the following result is true:
* If
is an ovoid in a ''finite'' projective space of dimension , then .
:(In the finite case, ovoids exist only in 3-dimensional spaces.)
*In a finite projective space of order (i.e. any line contains exactly points) and dimension any pointset
is an ovoid if and only if
and no three points are
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
(on a common line).
Replacing the word ''projective'' in the definition of an ovoid by ''affine'', gives the definition of an ''affine ovoid''.
If for an (projective) ovoid there is a suitable hyperplane
not intersecting it, one can call this hyperplane the ''hyperplane
at infinity'' and the ovoid becomes an affine ovoid in the affine space corresponding to
. Also, any affine ovoid can be considered a projective ovoid in the projective closure (adding a hyperplane at infinity) of the affine space.
Examples
In real projective space (inhomogeneous representation)
#
(hypersphere)
#
These two examples are ''quadrics'' and are projectively equivalent.
Simple examples, which are not quadrics can be obtained by the following constructions:
: (a) Glue one half of a hypersphere to a suitable hyperellipsoid in a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
way.
: (b) In the first two examples replace the expression by .
''Remark:'' The real examples can not be converted into the complex case (projective space over
). In a complex projective space of dimension there are no ovoidal quadrics, because in that case any non degenerated quadric contains lines.
But the following method guarantees many non quadric ovoids:
* For any ''non-finite'' projective space the existence of ovoids can be proven using ''transfinite induction''.
Finite examples
* Any ovoid
in a ''finite'' projective space of dimension over a field of
characteristic is a ''quadric''.
The last result can not be extended to even characteristic, because of the following non-quadric examples:
* For
odd and
the automorphism
the pointset
:
is an ovoid in the 3-dimensional projective space over (represented in inhomogeneous coordinates).
:Only when is the ovoid
a quadric.
:
is called the Tits-Suzuki-ovoid.
Criteria for an ovoid to be a quadric
An ovoidal quadric has many symmetries. In particular:
* Let be
an ovoid in a projective space
of dimension and
a hyperplane. If the ovoid is symmetric to any point
(i.e. there is an involutory perspectivity with center
which leaves
invariant), then
is pappian and
a quadric.
*An ovoid
in a projective space
is a quadric, if the group of projectivities, which leave
invariant operates 3-transitively on
, i.e. for two triples
there exists a projectivity
with
.
In the finite case one gets from
Segre's theorem
In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement:
*Any oval in a ''finite pappian'' projective plane of ''odd'' order is a nondegenerate projective conic section.
This statement was ...
:
* Let be
an ovoid in a ''finite'' 3-dimensional desarguesian projective space
of ''odd'' order, then
is pappian and
is a quadric.
Generalization: semi ovoid
Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid:
:A point set
of a projective space is called a ''semi-ovoid'' if
the following conditions hold:
:(SO1) For any point
the tangents through point
exactly cover a hyperplane.
: (SO2)
contains no lines.
A semi ovoid is a special ''semi-quadratic set'' which is a generalization of a ''
quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
''. The essential difference between a semi-quadratic set and a quadratic set is the fact, that there can be lines which have 3 points in common with the set and the lines are not contained in the set.
Examples of semi-ovoids are the sets of isotropic points of an
hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
. They are called ''hermitian quadrics''.
As for ovoids in literature there are criteria, which make a semi-ovoid to a hermitian quadric.
See, for example.
[K.J. Dienst: ''Kennzeichnung hermitescher Quadriken durch Spiegelungen'', Beiträge zur geometrischen Algebra (1977), Birkhäuser-Verlag, S. 83-85.]
Semi-ovoids are used in the construction of examples of Möbius geometries.
See also
*
Ovoid (polar space)
*
Möbius plane In mathematics, a Möbius plane (named after August Ferdinand Möbius) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane. The classical example is based on the geometry of lines and circles in the real affine plane.
A s ...
Notes
References
*
Further reading
*
*
*
External links
* E. Hartmann:
Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.' Skript, TH Darmstadt (PDF; 891 kB), S. 121-123.
{{DEFAULTSORT:Ovoid (Projective Geometry)
Projective geometry
Incidence geometry