# Ovoid (projective geometry)

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:
* Any line intersects ${\mathcal {O}}$ in at most 2 points,
* The tangents at a point cover a hyperplane (and nothing more), and
* ${\mathcal {O}}$ 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 ovoid

* In a projective space of dimension d ≥ 3 a set ${\mathcal {O}}$ of points is called an ovoid, if
    (1) Any line g meets ${\mathcal {O}}$ in at most 2 points.
In the case of $|g\cap {\mathcal {O}}|=0$, the line is called a passing (or exterior) line, if $|g\cap {\mathcal {O}}|=1$ the line is a tangent line, and if $|g\cap {\mathcal {O}}|=2$ the line is a secant line.
    (2) At any point $P\in {\mathcal {O}}$ the tangent lines through P cover a hyperplane, the tangent hyperplane, (i.e., a projective subspace of dimension d − 1).
    (3) ${\mathcal {O}}$ contains no lines.
From the viewpoint of the hyperplane sections, an ovoid is a rather homogeneous object, because
* For an ovoid ${\mathcal {O}}$ and a hyperplane $\varepsilon$, which contains at least two points of ${\mathcal {O}}$, the subset $\varepsilon \cap {\mathcal {O}}$ is an ovoid (or an oval, if 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:
* If ${\mathcal {O}}$ 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.)
* In a finite projective space of order 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 $|{\mathcal {O}}|=n^{2}+1$ 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 $\varepsilon _{\infty }$ at infinity and the ovoid becomes an affine ovoid in the affine space corresponding to $\varepsilon _{\infty }$. 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)

* ${\mathcal {O}}=\{(x_{1},...,x_{d})\in {\mathbb {R} }^{d}\;|\;x_{1}^{2}+\cdots +x_{d}^{2}=1\}\ ,$ (hypersphere)
* ${\mathcal {O}}=\{(x_{1},...,x_{d})\in {\mathbb {R} }^{d}\;|x_{d}=x_{1}^{2}+\cdots +x_{d-1}^{2}\;\}\;\cup \;\{{\text{point at infinity of }}x_{d}{\text{-axis}}\}$
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 x$_1$$^2$ by x$_1$$^4$.
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:
* For any non-finite projective space the existence of ovoids can be proven using transfinite induction.

### Finite examples

* Any ovoid ${\mathcal {O}}$ in a finite projective space of dimension 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:
* For $K=GF(2^{m}),\;m$ odd and $\sigma$ the automorphism $x\mapsto x^{(2^{\frac {m+1}{2}})}\;,$
the pointset
    ${\mathcal {O}}=\{(x,y,z)\in K^{3}\;|\;z=xy+x^{2}x^{\sigma }+y^{\sigma }\}\;\cup \;\{{\text{point of infinity of the }}z{\text{-axis}}\}$ is an ovoid in the 3-dimensional projective space over K (represented in inhomogeneous coordinates).
    Only when m = 1 is the ovoid ${\mathcal {O}}$ a quadric.
    ${\mathcal {O}}$ is called the Tits-Suzuki-ovoid.

## Criteria for an ovoid to be a quadric

An ovoidal quadric has many symmetries. In particular:
* Let be ${\mathcal {O}}$ an ovoid in a projective space ${\mathfrak {P}}$ of dimension d ≥ 3 and $\varepsilon$ a hyperplane. If the ovoid is symmetric to any point $P\in \varepsilon \setminus {\mathcal {O}}$ (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.
* An ovoid ${\mathcal {O}}$ in a projective space ${\mathfrak {P}}$ is a quadric, if the group of projectivities, which leave ${\mathcal {O}}$ invariant operates 3-transitively on ${\mathcal {O}}$, i.e. for two triples $A_{1},A_{2},A_{3},\;B_{1},B_{2},B_{3}$ there exists a projectivity $\pi$ with $\pi (A_{i})=B_{i},\;i=1,2,3$.
In the finite case one gets from Segre's theorem:
* Let be ${\mathcal {O}}$ an ovoid in a finite 3-dimensional desarguesian projective space ${\mathfrak {P}}$ of odd order, then ${\mathfrak {P}}$ is pappian and ${\mathcal {O}}$ 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 ${\mathcal {O}}$ of a projective space is called a semi-ovoid if
the following conditions hold:
    (SO1) For any point $P\in {\mathcal {O}}$ the tangents through point $P$ exactly cover a hyperplane.
    (SO2) ${\mathcal {O}}$ contains no lines.
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.