Oval (projective plane)
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In PG(2,q), with q a prime power, an oval is a set of <math>q+1</math> points, no three of which are collinear.
[edit] 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 :
<math>\{(t,t^2,1)\mid t\in GF(q)\}\cup \{(0,1,0)\}</math>
[edit] Even q
When <math>q=2^h</math>, the situation is completely different.
In this case, sets of <math>q+2</math> 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 :
<math>\{(t,t^2,1)\mid t\in GF(q)\}\cup \{(0,1,0)\}\cup\{(1,0,0)\}</math>
However, many other types of hyperovals can be found.
