---
tag: R3TMVHTP3A
kind: concept
name: Projective conic
msc: ["51N15", "51E21"]
sources:
  - url: https://en.wikipedia.org/wiki/Conic_section
---

## Summary

A projective conic is the set of points of a projective plane over a field where a quadratic form in three variables vanishes.
It is the projective version of the [[conic section|00HJCMC2CD]].
Ellipses, parabolas and hyperbolas become the same kind of curve once points at infinity are added.
Non-degenerate conics are the simplest curves after lines, and in finite planes they are the model examples of [[ovals|7B03TB64N9]].

## Explanations

### L1 human

An ellipse, a parabola and a hyperbola look different in the plane.
In the projective plane, a parabola gains one point at infinity and a hyperbola gains two, and all three become closed curves of the same type.
In homogeneous coordinates $[x : y : z]$ each one is given by an equation of degree $2$, such as $xz = y^2$.
This curve contains the points $[1 : t : t^2]$ for every $t$, and also the point $[0 : 0 : 1]$.