Plane curve

In mathematics, a plane curve is a curve in a plane that may be either a Euclidean 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. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. 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 at a point of the curve.

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 their projective completion are all isomorphic to the projective completion of the circle x² + y² = 1 (that is the projective curve of equation x² + y² - z²= 0). The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree four are called quartic plane 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.