---
tag: RNG2A8BFVZ
kind: concept
name: Hyperoval
msc: ["51E21", "51E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Oval_(projective_plane)
---

## Summary

A hyperoval in a finite projective plane of order $n$ is a set of $n + 2$ points with no three on a line.
It is the largest possible size of such a set, and it can occur only when $n$ is even.
When $n$ is even, every [[oval|7B03TB64N9]] extends to a hyperoval by adding one point, the nucleus, where all tangents of the oval meet.
The classification of hyperovals in $PG(2, 2^h)$ is a long-standing problem of finite geometry.

## Explanations

### L1 human

In a finite projective plane, how many points can one choose with no three on a line?
The answer is at most $n + 2$, and when that many can be chosen, the set is a hyperoval.
Every line then meets the set in $0$ or $2$ points, never in exactly one.
In the Fano plane, the four points off any line form a hyperoval.