Editing Plane curve

{{Short description|Mathematical concept}}
{{no footnotes|date=October 2018}}
In [[mathematics]], a '''plane curve''' is a [[curve]] in a [[plane (geometry)|plane]] that may be a [[Plane (mathematics)|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 curve]]s.
Plane curves also include the [[Jordan curve]]s (curves that enclose a region of the plane but need not be smooth) and the [[Graph of a function|graphs of continuous function]]s.

==Symbolic representation==
A plane curve can often be represented in [[Cartesian coordinates]] by an [[implicit equation]] of the form <math>f(x,y)=0</math> for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x''  – that is, rewritten as <math>y=g(x)</math> or <math>x=h(y)</math> 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 <math>(x,y)=(x(t), y(t))</math> for specific functions <math>x(t)</math> and <math>y(t).</math>

Plane curves can sometimes also be represented in alternative [[coordinate system]]s, 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==
A smooth plane curve is a curve in a [[real number|real]] Euclidean plane {{tmath|\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 (geometry)|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 <math>f(x, y) = 0,</math> where {{tmath|f: \R^2 \to \R}} is a [[smooth function]], and the [[partial derivative]]s {{tmath|\partial f/\partial x}} and {{tmath|\partial f/\partial y}} are never both 0 at a point of the curve.

==Algebraic plane curve==
An [[algebraic plane curve]] is a curve in an [[affine plane|affine]] or [[projective plane]] given by one polynomial equation <math>f(x,y) = 0</math> (or <math>F(x,y,z) = 0,</math> where {{mvar|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 a polynomial|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 <math>x^2 + y^2 = 1</math> has degree 2.

The [[Algebraic curve#Singularities|non-singular]] plane algebraic curves of degree 2 are called [[conic section]]s, and their [[projective completion]] are all [[isomorphic]] to the projective completion of the circle <math>x^2 + y^2 = 1</math> (that is the projective curve of equation {{nowrap|<math>x^2 + y^2 - z^2 = 0</math>).}} The plane curves of degree 3 are called [[cubic plane curve]]s and, if they are non-singular, [[elliptic curve]]s. Those of degree 4 are called [[quartic plane curve]]s.

==Examples==

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):

{| class="wikitable"
|-
! Name
! [[Implicit equation]]
! [[Parametric equation]]
! As a [[function (mathematics)|function]]
! graph
|-
| [[Line (geometry)|Straight line]]
| <math>a x+b y=c</math>
| <math>(x,y)=(x_0 + \alpha t,y_0+\beta t)</math>
| <math>y=m x+c</math>
| [[File:Gerade.svg|frameless|100px]]
|-
| [[Circle]]
| <math>x^2+y^2=r^2</math>
| <math>(x,y)=(r \cos t, r \sin t)</math>
|
| [[File:Centre de gravite disque.svg|framless|100px]]
|-
| [[Parabola]]
| <math>y-x^2=0</math>
| <math>(x,y)=(t,t^2)</math>
| <math>y=x^2</math>
| [[File:Parabola.svg|frameless|100px]]
|-
| [[Ellipse]]
| <math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math>
| <math>(x,y)=(a \cos t, b \sin t)</math>
| 
| [[File:Simple Ellipse.svg|framless|100px]]
|-
| [[Hyperbola]]
| <math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1</math>
| <math>(x,y)=(a \cosh t, b \sinh t)</math>
|
| [[File:Hyperbola.svg|frameless|100px]]
|}

== See also ==
* [[Algebraic geometry]]
* [[Convex curve]]
* [[Differential geometry]]
* [[Osgood curve]]
* [[Plane curve fitting]]
* [[Projective varieties]]
* [[Skew curve]]

== References ==
*{{Citation|first=J. L.|last=Coolidge|title=A Treatise on Algebraic Plane Curves|publisher=Dover Publications|date=April 28, 2004|ISBN=0-486-49576-0}}.
*{{Citation|first=R. C.|last=Yates|title=A handbook on curves and their properties|publisher=J.W. Edwards|year=1952|asin=B0007EKXV0}}.
*{{Citation|first=J. Dennis|last=Lawrence|title=A catalog of special plane curves|publisher=Dover|year=1972|ISBN=0-486-60288-5|url-access=registration|url=https://archive.org/details/catalogofspecial00lawr}}.

== External links ==
* {{MathWorld |id=PlaneCurve |title=Plane Curve}}

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[[Category:Euclidean geometry]]
[[Category:Plane curves| ]]

[[es:Curva plana]]