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 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
Straight line	$a x+b y=c$	$(x_0 + \alpha t,y_0+\beta t)$	$y=m x+c$
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	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$(a \cosh t, b \sinh t)$

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.

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form

Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm

Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law

Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve

Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem

Singularities	Acnode Crunode Cusp Delta invariant Tacnode

Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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