---
tag: EWS4WFTA7W
kind: concept
name: Plane curve
msc: [14H50]
specializes: [XJM2MT4S3Y]
sources:
  - url: https://en.wikipedia.org/wiki/Plane_curve
---

## Summary

A plane algebraic curve is a curve in the affine or projective plane defined by one polynomial equation.
Its degree is the degree of that polynomial.
By [[Bézout's theorem|KWNVF194D8]], it is the number of points in which a general line meets the curve.
Lines, conics and cubics are the plane curves of degree $1$, $2$ and $3$.
More generally, in geometry a plane curve is any curve that lies in a plane.

## Explanations

### L1 human

A plane curve is the set of points $(x, y)$ satisfying one polynomial equation $f(x, y) = 0$.
The circle $x^2 + y^2 = 1$ has degree $2$, and the curve $y^2 = x^3 - x$ has degree $3$.
Adding points at infinity gives a projective plane curve, defined by a homogeneous polynomial in three variables.
Many properties of the curve, such as its [[genus|6W8TDV9VVN]] when it is smooth, can be read off from the degree alone.