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''ovaloid''

A set  $  O $
of points in some space which is intersected by an arbitrary straight line in at most two points, and such that the tangents to  $  O $
at each of its points cover exactly a hyperplane. In projective space a non-ruled [[Quadric|quadric]] is an ovoid. This term is mainly used in finite geometries.

In finite projective spaces of dimension greater than three, ovoids do not exist. In three-dimensional spaces of order  $  q > 2 $, 
an ovoid is a maximal  $  k $-
cap (cf. [[Cap|Cap]]) and consists of  $  q  ^ {2} + 1 $
points, and for odd  $  q $
any ovoid is an elliptic quadric (see [[#References|[1]]]). In a plane of order  $  q $, 
an ovoid is called an oval, and consists of  $  q + 1 $
points. In a Desarguesian plane of odd order, any oval is uniquely representable as a non-degenerate conic over a Galois field (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Segre,   "Introduction to Galois geometries"  ''Atti Accad. Naz. Lincei'' , '''8'''  (1967)  pp. 133–236</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B. Segre,   "Ovals in a finite projective plane"  ''Canad. J. Math.'' , '''7'''  (1955)  pp. 414–416</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Tits,   "Ovoids à translations"  ''Rend. Mat. e Appl.'' , '''21'''  (1962)  pp. 37–59</TD></TR></table>

====Comments====
For Desarguesian planes of even order there are counterexamples to the last statement above.

An ovoid in  $  \mathbf P  ^ {3} $
is a set  $  {\mathcal O} $
of points such that no four lie in a plane and such that at each  $  A \in {\mathcal O} $
there is a unique hyperplane through  $  A $
tangent to  $  {\mathcal O} $
at that point. Here  "tangent"  means that the intersection of  $  {\mathcal O} $
with the hyperplane consists only of  $  A $
itself.

For a finite field of odd characteristic the ovoids in  $  \mathbf P  ^ {3} $
are precisely the zeros of a quadratic form of Witt index 1, [[#References|[a1]]].

An ovoid in a [[Polar space|polar space]] (in particular, in a generalized [[Quadrangle|quadrangle]]) is a collection  $  {\mathcal O} $
of points such that every maximal singular subspace intersects  $  {\mathcal O} $
in exactly one point. A spread in a generalized quadrangle is a set  $  {\mathcal R} $
of lines such that each point is incident with one line of  $  {\mathcal R} $. 
A spread is an ovoid in the dual generalized quadrangle. An ovoid in a finite generalized quadrangle of order  $  ( s, t) $
has cardinality  $  st+ 1 $.

A (trivial) example of an ovoid is the set of encircled points in the grid (cf. [[Quadrangle|Quadrangle]]) depicted below:

[[File:Ovoid grid.svg|300px|center|Ovoid]]

The connection between the abstract notion of an ovoid in a polar space and an ovoid in  $  \mathbf P  ^ {3} $
is as follows. Consider the classical generalized quadrangle defined by a symplectic bilinear form  $  Q $. 
I.e. the points are the points of  $  \mathbf P  ^ {3} $(
which are all isotropic) and the lines are the totally isotropic lines of  $  \mathbf P  ^ {3} $
with respect to this form. Then an ovoid in this generalized quadrangle viewed as a subset of  $  \mathbf P  ^ {3} $
is an ovoid in the sense of the geometric version of the concept. (The tangent plane to  $  ( y _ {0} :  y _ {1} :  y _ {2} :  y _ {3} ) = A \in {\mathcal O} $
is  $  A  ^  \perp  = \{ {x \in \mathbf P  ^ {3} } : {Q( x, y) = 0 } \} $.

Let  $  \Omega  ^ {+} ( 2n, \mathbf F ) $, 
$  \mathbf F $
a finite field, be the (classical) polar space defined by the bilinear form

$$ 
x _ {0} x _ {1} + x _ {2} x _ {3} + \dots + x _ {2n-} 2 x _ {2n-} 1 .
$$

Ovoids in  $  \Omega  ^ {+} ( 6, \mathbf F ) $
are used to obtain non-Desarguesian translation planes. From one  "master"  ovoid in  $  \Omega  ^ {+} ( 8, \mathbf F ) $
one obtains many ovoids in  $  \Omega  ^ {+} ( 6, \mathbf F ) $. 
It is an open problem whether there are ovoids in  $  \Omega  ^ {+} ( 10, \mathbf F ) $. 
There are none  $  \Omega  ^ {+} ( 10, \mathbf F _ {3} ) $, 
[[#References|[a4]]], or in  $  \Omega  ^ {+} ( 10 , \mathbf F _ {2} ) $, 
[[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Barlotti,   "Un' estenzione del teorema di Segre–Kustaanheimo"  ''Boll. Un. Mat. Ital. (3)'' , '''10'''  (1955)  pp. 498–506</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.E. Pagne,   J.A. Thas,   "Finite generalized quadrangles" , Pitman  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Mason,   E.E. Shult,   "The Klein correspondence and the ubiquity of certain translation planes"  ''Geom. Dedicata'' , '''21'''  (1986)  pp. 29–50</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.E. Shult,   "Nonexistence of ovoids in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070670/o07067039.png" />"  ''J. Comb. Theory, Ser. A'' , '''51'''  (1989)  pp. 250–257</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.M. Kantor,   "Ovoids and translation planes"  ''Canad. J. Math.'' , '''34'''  (1982)  pp. 1195–1207</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.W.P. Hirschfeld,   "Finite projective spaces of three dimensions" , Clarendon Press  (1985)  pp. Chapt. 16</TD></TR></table>