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 \mathcal O are: 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 ovoid

d ≥ 3 a set \mathcal O of points is called an ovoid, if 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. d − 1 ). From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because d = 3 ) within the hyperplane \varepsilon. For finite projective spaces of dimension d ≥ 3 (i.e., the point set is finite, the space is pappian ), the following result is true: d ≥ 3 , then d = 3 . n >2 (i.e. any line contains exactly n + 1 points) and dimension d = 3 any pointset \mathcal O is an ovoid if and only if and no three points are collinear (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 \varepsilon 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)

These two examples are quadrics and are projectively equivalent. Simple examples, which are not quadrics can be obtained by the following constructions: x12 by x14 . Remark: The real examples can not be converted into the complex case (projective space over {\mathbb C}). 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:

Finite examples

d = 3 over a field K of characteristic ≠ 2 is a quadric. The last result can not be extended to even characteristic, because of the following non-quadric examples: the pointset m = 1 is the ovoid \mathcal O a quadric.

Criteria for an ovoid to be a quadric

An ovoidal quadric has many symmetries. In particular: d ≥ 3 and \varepsilon a hyperplane. If the ovoid is symmetric to any point (i.e. there is an involutory perspectivity with center P which leaves \mathcal O invariant), then \mathfrak P is pappian and \mathcal O a quadric. In the finite case one gets from Segre's theorem:

Generalization: semi ovoid

Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid: the following conditions hold: A semi ovoid is a special semi-quadratic set 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. Semi-ovoids are used in the construction of examples of Möbius geometries.