---
tag: HNTE5FPS14
kind: concept
name: Ovoid
msc: ["51E20", "51E21"]
sources:
  - url: https://en.wikipedia.org/wiki/Ovoid_(projective_geometry)
same_as:
  wikidata: Q7114196
  wikipedia: Ovoid (projective geometry)
---

## Summary

An ovoid is a set of points of a [[projective space|ZT9A6JK8QF]] that looks like a sphere from the point of view of incidence.
Every line meets it in at most two points, and at each of its points the tangent lines fill a hyperplane.
Ovoids in three-dimensional space are the spatial analogue of [[ovals|7B03TB64N9]] in a plane.
They give Möbius planes and generalized quadrangles, and finite ovoids are a main topic of Galois geometry.

## Explanations

### L1 human

Think of a sphere in space, completed with points at infinity.
A line pierces a sphere in at most two points, and the lines that just touch the sphere at a point make up its tangent plane.
An ovoid is any set of points with these two properties, defined only through points, lines and planes.
Over a finite field with $q$ elements, an ovoid of three-dimensional projective space has exactly $q^2 + 1$ points.