Oval (projective plane)

In PG(2,q), with q a prime power, an oval is a set of $q+1$ points, no three of which are collinear.

Odd q

When q is an odd prime power, no sets with more points than q+1, no three of which collinear, exist.

Due to Segre's theorem, every oval in PG(2,q) with q odd, is projective equivalent with a nonsingular conic in the plane.

This implies that a basis exists for every oval such that it has this parametrization :

$\{(t,t^{2},1)\mid t\in GF(q)\}\cup \{(0,1,0)\}$

Even q

When $q=2^{h}$, the situation is completely different.

In this case, sets of $q+2$ points, no three of which collinear, exist and they are called hyperovals.

When q is even, one can show that there is a unique tangent through each point, and that all these tangents are concurrent in a point p outside the oval. Adding this point to the oval gives a hyperoval. Conversely, removing one point from a hyperoval immediately gives an oval.

Every nonsingular conic in the projective plane, together with its kernel, forms a hyperoval. For each of these sets, there is a basis such that the set is :

$\{(t,t^{2},1)\mid t\in GF(q)\}\cup \{(0,1,0)\}\cup \{(1,0,0)\}$

However, many other types of hyperovals can be found.

See also

• [Ovoid (projective geometry)]