Oval (projective plane)

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In PG(2,q), with q a prime power, an oval is a set of $q + 1$ points, no three of which are collinear. Indeed, in any finite projective plane of order q, an oval is a set of $q + 1$ points, no three collinear.

[edit] Odd q

In an finite projective plane of odd order q, no sets with more points than q+1, no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to statistical design of experiments.

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

This implies that, after a possible change of cordinates, every oval of PG(2,q) with q odd has the parametrization :

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

[edit] Even q

When $q$ is even, the situation is completely different.

In this case, sets of $q + 2$ points, no three of which collinear, may exist in a finite projective plane of order q and they are called hyperovals.

When q is even, given an oval, Qvist in 1952 showed 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 (called the nucleus of the oval) 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 nucleus, forms a hyperoval. For each of these sets, there is a system of coordinates 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 of PG(2,q) can be found if $q > 8$ . Hyperovals of PG(2,q) for q even have only been classified for $q < 64$ to date.