Editing Plane curve (section)

==Algebraic plane curve==
An [[algebraic plane curve]] is a curve in an [[affine plane|affine]] or [[projective plane]] given by one polynomial equation <math>f(x,y) = 0</math> (or <math>F(x,y,z) = 0,</math> where {{mvar|F}} is a [[homogeneous polynomial]], in the projective case.)

Algebraic curves have been studied extensively since the 18th century.

Every algebraic plane curve has a degree, the [[degree of a polynomial|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 <math>x^2 + y^2 = 1</math> has degree 2.

The [[Algebraic curve#Singularities|non-singular]] plane algebraic curves of degree 2 are called [[conic section]]s, and their [[projective completion]] are all [[isomorphic]] to the projective completion of the circle <math>x^2 + y^2 = 1</math> (that is the projective curve of equation {{nowrap|<math>x^2 + y^2 - z^2 = 0</math>).}} The plane curves of degree 3 are called [[cubic plane curve]]s and, if they are non-singular, [[elliptic curve]]s. Those of degree 4 are called [[quartic plane curve]]s.