Plane curve

In mathematics, a plane curve is a curve in a plane, that may be either aEuclidean plane, an affine plane or aprojective 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 R² and is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x, y) = 0, where f : R² → 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 x² + y² = 1 has degree 2.

The non-singular plane algebraic curves of degree 2 are called conic sections, and are isomorphic to the projective completion of the circle x² + y² = 1 (that is the projective curve of equation x² + y² - z²= 0). The non-singular plane curves of degree 3 are called elliptic curves.

Examples

Name	Implicit equation	Parametric equation	As a function	graph
Straight line	$ax+by=c$	$(x_{0}+\alpha t,y_{0}+\beta t)$	$y=mx+c$
Circle	$x^{2}+y^{2}=r^{2}$	$(r\cos t,r\sin t)$
Parabola	$y-x^{2}=0$	$(t,t^{2})$	$y=x^{2}$
Ellipse	${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$	$(a\cos t,b\sin t)$
Hyperbola	${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$	$(a\cosh t,b\sinh t)$

See also

• Algebraic curve
• Differential geometry
• Algebraic geometry
• Plane curve fitting
• Projective varieties
• Two-dimensional graph

References

• Coolidge, J. L. (April 28, 2004), A Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0-486-49576-0.
• Yates, R. C. (1952), A handbook on curves and their properties, J.W. Edwards, ASIN B0007EKXV0.

External links

• Weisstein, Eric W. "Plane Curve". MathWorld.