Plane curve

In

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

A plane curve can often be represented in

implicit equation

of the form

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)

or

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))

for specific functions

x(t)

and

y(t).

Plane curves can sometimes also be represented in alternative

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 Euclidean plane $\mathbb {R} ^{2}$ and is a one-dimensional

smooth function

. Equivalently, a smooth plane curve can be given locally by an equation

f(x,y)=0,

where

f:\mathbb {R} ^{2}\to \mathbb {R}

is a

smooth function, and the partial derivatives

\partial f/\partial x

and

\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 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 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 $x^{2}+y^{2}=1$ has degree 2.

The

isomorphic

to the projective completion of the circle

x^{2}+y^{2}=1

(that is the projective curve of equation

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

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