Ovoid (projective geometry)

To the definition of an ovoid: t tangent, s secant line

In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid are:

  1. Any line intersects in at most 2 points,
  2. The tangents at a point cover a hyperplane (and nothing more), and
  3. 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 in a projective plane.

An ovoid is a special type of a quadratic set.

Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries.

Definition of an ovoidEdit

  • In a projective space of dimension d ≥ 3 a set   of points is called an ovoid, if
(1) Any line g 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 P cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension d − 1).
(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 d = 3) within the hyperplane  .

For finite projective spaces of dimension d ≥ 3 (i.e., the point set is finite, the space is pappian[1]), the following result is true:

  • If   is an ovoid in a finite projective space of dimension d ≥ 3, then d = 3.
(In the finite case, ovoids exist only in 3-dimensional spaces.)[2]
  • In a finite projective space of order n >2 (i.e. any line contains exactly n + 1 points) and dimension d = 3 any pointset   is an ovoid if and only if   and no three points are collinear (on a common line).[3]

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.


In real projective space (inhomogeneous representation)Edit

  1.   (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 way.
(b) In the first two examples replace the expression x12 by x14.

Remark: The real examples can not be converted into the complex case (projective space over  ). In a complex projective space of dimension d ≥ 3 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.[4][5]

Finite examplesEdit

  • Any ovoid   in a finite projective space of dimension d = 3 over a field K of characteristic ≠ 2 is a quadric.[6]

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 K (represented in inhomogeneous coordinates).
Only when m = 1 is the ovoid   a quadric.[7]
  is called the Tits-Suzuki-ovoid.

Criteria for an ovoid to be a quadricEdit

An ovoidal quadric has many symmetries. In particular:

  • Let be   an ovoid in a projective space   of dimension d ≥ 3 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.[8]
  • 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  .[9]

In the finite case one gets from Segre's theorem:

  • Let be   an ovoid in a finite 3-dimensional desarguesian projective space   of odd order, then   is pappian and   is a quadric.

Generalization: semi ovoidEdit

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[10] which is a generalization of a quadratic set. 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. 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[11].

Semi-ovoids are used in the construction of examples of Möbius geometries.

See alsoEdit


  1. ^ Dembowski 1968, p. 28
  2. ^ Dembowski 1968, p. 48
  3. ^ Dembowski 1968, p. 48
  4. ^ W. Heise: Bericht über  -affine Geometrien, Journ. of Geometry 1 (1971), S. 197–224, Satz 3.4.
  5. ^ F. Buekenhout: A Characterization of Semi Quadrics, Atti dei Convegni Lincei 17 (1976), S. 393-421, chapter 3.5
  6. ^ Dembowski 1968, p. 49
  7. ^ Dembowski 1968, p. 52
  8. ^ H. Mäurer: Ovoide mit Symmetrien an den Punkten einer Hyperebene, Abh. Math. Sem. Hamburg 45 (1976), S.237-244
  9. ^ J. Tits: Ovoides à Translations, Rend. Mat. 21 (1962), S. 37–59.
  10. ^ F. Buekenhout: A Characterization of Semi Quadrics, Atti dei Convegni Lincei 17 (1976), S. 393-421.
  11. ^ K.J. Dienst: Kennzeichnung hermitescher Quadriken durch Spiegelungen, Beiträge zur geometrischen Algebra (1977), Birkhäuser-Verlag, S. 83-85.


  • Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275

Further readingEdit

  • Barlotti, A. (1955), "Un'estensione del teorema di Segre-Kustaanheimo", Boll. Un. Mat. Ital., 10: 96–98
  • Hirschfeld, J.W.P. (1985), Finite Projective Spaces of Three Dimensions, New York: Oxford University Press, ISBN 0-19-853536-8
  • Panella, G. (1955), "Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito", Boll. Un. Mat. Ital., 10: 507–513

External linksEdit