---
tag: Q3Q4FF23VG
kind: concept
name: Quadric
msc: ["51N15", "51A50", "14J70"]
sources:
  - url: https://en.wikipedia.org/wiki/Quadric
same_as:
  wikidata: Q852117
  wikipedia: Quadric
---

## Summary

A quadric is the set of points of an affine or projective space where a polynomial of degree $2$ vanishes.
Conics in the plane and quadric surfaces such as spheres, ellipsoids and hyperboloids in space are quadrics.
In projective geometry, a quadric is given by a [[quadratic form|0RBQS1JJCR]].
The points, lines and subspaces lying on a non-degenerate quadric form a [[polar space|Z7T0V2VAWJ]].

## Explanations

### L1 human

A quadric is the next simplest shape after a flat one: flat sets are cut out by equations of degree $1$, quadrics by equations of degree $2$.
The sphere $x^2 + y^2 + z^2 = 1$ is a quadric in space.
So is the saddle $z = xy$, which contains two families of straight lines.
In the projective setting all equations are written homogeneously, for example $x_0 x_1 + x_2 x_3 = 0$.