# Plane curve

In mathematics, a **plane curve** is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.

## Smooth plane curve

A smooth plane curve is a curve in a real Euclidean plane **R**^{2} and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function.
Equivalently, a smooth plane curve can be given locally by an equation *f*(*x*, *y*) = 0, where *f* : **R**^{2} → **R** is a smooth function, and the partial derivatives ∂*f*/∂*x* and ∂*f*/∂*y* are never both 0 at a point of the curve.

## Algebraic plane curve

An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation *f*(*x*, *y*) = 0 (or *F*(*x*, *y*, *z*) = 0, where *F* is a homogeneous polynomial, in the projective case.)

Algebraic curves were studied extensively since the 18th century.

Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation *x*^{2} + *y*^{2} = 1 has degree 2.

The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle *x*^{2} + *y*^{2} = 1 (that is the projective curve of equation *x*^{2} + *y*^{2} - *z*^{2}= 0). The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree four are called quartic plane curves.

## Examples

Name | Implicit equation | Parametric equation | As a function | graph |
---|---|---|---|---|

Straight line | ||||

Circle | ||||

Parabola | ||||

Ellipse | ||||

Hyperbola |

## See also

- Algebraic curve
- Differential geometry
- Algebraic geometry
- Plane curve fitting
- Projective varieties
- Two-dimensional graph

## References

- Coolidge, J. L. (April 28, 2004),
*A Treatise on Algebraic Plane Curves*, Dover Publications, ISBN 0-486-49576-0. - Yates, R. C. (1952),
*A handbook on curves and their properties*, J.W. Edwards, ASIN B0007EKXV0.