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Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane (compare with 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 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. 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 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, 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.
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).
Contents
Examples
Name  Implicit equation  Parametric equation  As a function  graph 

Straight line  <math>a x+b y=c</math>  <math>(x_0 + \alpha t,y_0+\beta t)</math>  <math>y=m x+c</math>  100px 
Circle  <math>x^2+y^2=r^2</math>  <math>(r \cos t, r \sin t)</math>  100px  
Parabola  <math>yx^2=0</math>  <math>(t,t^2)</math>  <math>y=x^2</math>  100px 
Ellipse  <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math>  <math>(a \cos t, b \sin t)</math>  100px  
Hyperbola  <math>\frac{x^2}{a^2}  \frac{y^2}{b^2} = 1</math>  <math>(a \cosh t, b \sinh t)</math>  100px 
See also
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

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