---
tag: 7B03TB64N9
kind: concept
name: Oval
msc: ["51E21", "51E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Oval_(projective_plane)
same_as:
  wikidata: Q7113209
  wikipedia: Oval (projective plane)
---

## Summary

An oval in a projective plane is a set of points that behaves like a conic.
No line meets it in more than two points, and at each of its points there is exactly one tangent line.
In a finite plane of order $n$, an oval is the same as a set of $n + 1$ points with no three on a line.
Non-degenerate [[projective conics|R3TMVHTP3A]] are ovals.
[[Segre's theorem|SPFP3GHVYJ]] says that in $PG(2, q)$ with $q$ odd there are no others.
For $q$ even, ovals extend to [[hyperovals|RNG2A8BFVZ]], and for $q \ge 8$ there are ovals that are not conics.

## Explanations

### L1 human

A circle meets any line in at most two points, and at each of its points it has a single tangent.
An oval keeps these two properties, stated only in terms of points and lines, so it makes sense in any projective plane.
In the Fano plane, the four points off a fixed line form a set with no three on a line; any three of them form an oval.
Ovals are the starting point for the study of arcs and caps in finite geometry.