# Quadratic set

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In mathematics, a **quadratic set** is a set of points in a projective plane/space which 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 quadratic set

Let be a projective space. A non empty subset of is called **quadratic set** if

**(QS1)**Any line of intersects in at most 2 points or is contained in .- ( is called
**exterior, tangent**and**secant**line if and respectively.) **(QS2)**For any point the union of all tangent lines through is a hyperplane or the entire space .

A quadratic set is called **non degenerated** if for any point set
is a hyperplane.

The following result is an astonishing statement for finite projective spaces.

**Theorem(BUEKENHOUT):**
Let be a *finite* projective space of dimension and
a non degenerated quadratic set which contains lines. Then:
is pappian and is a *quadric* with index .

## Definition of an oval and an ovoid

Ovals and ovoids are special quadratic sets:

Let be a projective space of dimension . A non degenerated quadratic set that does not contain lines is called **ovoid** (or **oval** in plane case).

The following equivalent definition of an oval/ovoid are more common:

**Definition: (oval)**
A non empty point set of a projective plane is called
**oval** if the following properties are fulfilled:

**(o1)**Any line meets in at most two points.- (
**o2)**For any point there is one and only one line such that .

A line is a *exterior* or *tangent* or *secant* line of the
oval if \ or or respectively.

For *finite* planes the following theorem provides a more simple definition.

**Theorem: (oval in finite plane) **Let be a projective plane of order .
A set of points is an **oval** if and if no three points
of are collinear.

For *pappian* projective planes of *odd* order the ovals are just conics:

**Theorem (SEGRE):**
Let be a *pappian* projective plane of *odd* order.
Any oval in is an oval *conic* (non degenerate quadric).

**Definition: (ovoid)**
A non empty point set of a projective space is called **ovoid** if the following properties are fulfilled:

**(O1)**Any line meets in at most two points.- ( is called
**exterior, tangent**and**secant**line if and respectively.) **(O2)**For any point the union of all tangent lines through is a hyperplane (tangent plane at ).

**Example:**

- a) Any sphere (quadric of index 1) is an ovoid.
- b) In case of real projective spaces one can construct ovoids by combining halves of suitable ellipsoids such that they are no quadrics.

For *finite* projective spaces of dimension over a field we have:

**Theorem:**

Counter examples (TITS–SUZUKI-ovoid) show that i.g. statement b) of the theorem above is not true for :

## External links

- Lecture Note
, an Introduction to Moebius-, Laguerre- and Minkowski Planes, p. 121**Planar Circle Geometries**

## References

- F. Buekenhout (ed.),
*Handbook of Incidence Geometry*, Elsevier (1995) ISBN 0-444-88355-X - P. Dembowski,
**Finite Geometries**, Springer-Verlag (1968) ISBN 3-540-61786-8, p. 48