Plane curve
In mathematics, a plane curve is a curve in a plane, that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
Smooth plane curve
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.
Algebraic plane curve
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 since the 18th century.
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.
The nonsingular plane algebraic curves of degree 2 are called conic sections, and are isomorphic to the of the circle x^{2} + y^{2} = 1 (that is the projective curve of equation x^{2} + y^{2}  z^{2}= 0). The nonsingular plane curves of degree 3 are called elliptic curves, and those of degree four are called quartic plane curves.
Examples
Name  Implicit equation  Parametric equation  As a function  graph 

Straight line  
Circle  
Parabola  
Ellipse  
Hyperbola 
See also
 Algebraic curve
 Differential geometry
 Algebraic geometry
 Plane curve fitting
 Projective varieties
 Twodimensional graph
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", MathWorld.

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