Plane curve
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In mathematics, a plane curve is a curve in an Euclidian plane. (Compare with the notion of a space curve.)
The most frequently studied cases are smooth plane curves (and piece-wise smooth plane curves), and algebraic plane curves.
Smooth plane curve is a curve in a real euclidian plane R2 with is an one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x,y) = 0, where f is a smooth function of two variables, and the partial derivatives fx and fy are not simoultaneously equal to 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 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 in 18-20s century, leading to a very rich and deep theory. The founders of the theory are I. Newton, B. Riemann et al, some main contributors are Abel, A. Poincare, M. Noether, et al. 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, a circle 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 projectivisation 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].
There are many questions in the theory of plane algebraic curves for which the answer is not known as of the beginning of the 21st century.
See also: Smooth manifolds
Some references:
1. Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, 1959.
2. Yates, R. C. A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, 1947.
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