Plane Curve

In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.

A smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation ƒ(x,y) = 0, where ƒ : R2 → R is a smooth function, and the partial derivatives ∂ƒ/∂x and ∂ƒ/∂y are never both 0. In other words, a smooth plane curve is a plane curve which "locally looks like a line" with respect to a smooth change of coordinates.

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

Algebraic curves were studied extensively in the 18th to 20th centuries, leading to a very rich and deep theory. Some founders of the theory are considered to be Isaac Newton and Bernhard Riemann, with main contributors being Niels Henrik Abel, Henri Poincaré, Max Noether, among others. Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line. For example, the circle given by the equation x2 + y2 = 1 has degree 2.

An important classical result states that every non-singular plane curve of degree 2 in a projective plane is isomorphic to the projection of the circle x2 + y2 = 1. However, the theory of plane curves of degree 3 is already very deep, and connected with the Weierstrass's theory of bi-periodic complex analytic functions (cf. elliptic curves, Weierstrass P-function).