 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 R^{2} and is a onedimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation ƒ(x,y) = 0, where ƒ : R^{2} → 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 x^{2} + y^{2} = 1 has degree 2.
An important classical result states that every nonsingular plane curve of degree 2 in a projective plane is isomorphic to the projection of the circle x^{2} + y^{2} = 1. However, the theory of plane curves of degree 3 is already very deep, and connected with the Weierstrass's theory of biperiodic complex analytic functions (cf. elliptic curves, Weierstrass Pfunction).
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
 Differential geometry
 Algebraic curve
 Algebraic geometry
 Projective varieties
References
 Coolidge, J. L. (April 28, 2004), A Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0486495760.
 Yates, R. C. (1952), A handbook on curves and their properties, J.W. Edwards, ASIN B0007EKXV0.
External links
 Weisstein, Eric W., "Plane Curve" from MathWorld.
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