Plane curve

Mathematical concept

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In mathematics, a plane curve is a curve in a plane that may be 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. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions.

## Symbolic representation

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A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form  f ( x , y ) = 0 {\displaystyle f(x,y)=0} for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as  y = g ( x ) {\displaystyle y=g(x)} or  x = h ( y ) {\displaystyle x=h(y)} for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form  ( x , y ) = ( x ( t ) , y ( t ) ) {\displaystyle (x,y)=(x(t),y(t))} for specific functions  x ( t ) {\displaystyle x(t)} and  y ( t ) . {\displaystyle y(t).}

Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.

## Smooth plane curve

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A smooth plane curve is a curve in a real Euclidean plane ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠ 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 , {\displaystyle f(x,y)=0,} where ⁠ f : R 2 -> R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } ⁠ is a smooth function, and the partial derivatives ⁠ ∂ f / ∂ x {\displaystyle \partial f/\partial x} ⁠ and ⁠ ∂ f / ∂ y {\displaystyle \partial f/\partial y} ⁠ are never both 0 at a point of the curve.

## Algebraic plane curve

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An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation  f ( x , y ) = 0 {\displaystyle f(x,y)=0} (or  F ( x , y , z ) = 0 , {\displaystyle F(x,y,z)=0,} where F is a homogeneous polynomial, in the projective case.)

Algebraic curves have been 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 {\displaystyle x^{2}+y^{2}=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  x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} (that is the projective curve of equation  x 2 + y 2 − z 2 = 0 {\displaystyle x^{2}+y^{2}-z^{2}=0} ). The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.

## Examples

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Numerous examples of plane curves are shown in Gallery of curves and listed at List of curves. The algebraic curves of degree 1 or 2 are shown here (an algebraic curve of degree less than 3 is always contained in a plane):

Name  | Implicit equation | Parametric equation | As a function | graph

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Straight line |  a x + b y = c {\displaystyle ax+by=c} |  ( x , y ) = ( x 0 + α t , y 0 + β t ) {\displaystyle (x,y)=(x_{0}+\alpha t,y_{0}+\beta t)} |  y = m x + c {\displaystyle y=mx+c} |
Circle |  x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} |  ( x , y ) = ( r cos ⁡ t , r sin ⁡ t ) {\displaystyle (x,y)=(r\cos t,r\sin t)} |  |
Parabola |  y − x 2 = 0 {\displaystyle y-x^{2}=0} |  ( x , y ) = ( t , t 2 ) {\displaystyle (x,y)=(t,t^{2})} |  y = x 2 {\displaystyle y=x^{2}} |
Ellipse |  x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1} |  ( x , y ) = ( a cos ⁡ t , b sin ⁡ t ) {\displaystyle (x,y)=(a\cos t,b\sin t)} |  |
Hyperbola |  x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1} |  ( x , y ) = ( a cosh ⁡ t , b sinh ⁡ t ) {\displaystyle (x,y)=(a\cosh t,b\sinh t)} |  |

## See also

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* Algebraic geometry
* Convex curve
* Differential geometry
* Osgood curve
* Plane curve fitting
* Projective varieties
* Skew curve

## References

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* 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.
* Lawrence, J. Dennis (1972), A catalog of special plane curves, Dover, ISBN 0-486-60288-5.

## External links

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

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Topics in algebraic curves

Rational curves|

* Five points determine a conic
* Projective line
* Rational normal curve
* Riemann sphere
* Twisted cubic

Elliptic curves|

| Analytic theory|

* Elliptic function
* Elliptic integral
* Fundamental pair of periods
* Modular form

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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|

* De Franchis theorem
* Faltings's theorem
* Hurwitz's automorphisms theorem
* Hurwitz surface
* Hyperelliptic curve

Plane curves|

* AF+BG theorem
* Bézout's theorem
* Bitangent
* Cayley–Bacharach theorem
* Conic section
* Cramer's paradox
* Cubic plane curve
* Fermat curve
* Genus–degree formula
* Hilbert's sixteenth problem
* Nagata's conjecture on curves
* Plücker formula
* Quartic plane curve
* Real plane curve

Riemann surfaces|

* Belyi's theorem
* Bring's curve
* Bolza surface
* Compact Riemann surface
* Dessin d'enfant
* Differential of the first kind
* Klein quartic
* Riemann's existence theorem
* Riemann–Roch theorem
* Teichmüller space
* Torelli theorem

Constructions|

* Dual curve
* Polar curve
* Smooth completion

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

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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 (Algebraic curves)

Singularities|

* Ak singularity
* Acnode
* Crunode
* Cusp
* Delta invariant
* Tacnode

Vector bundles|

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

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