Plane curve

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

A plane curve can often be represented in Cartesian coordinates by an 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 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

A smooth plane curve is a curve in a real Euclidean plane ⁠ $\mathbb {R} ^{2}$ ⁠ 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 $f(x,y)=0,$ 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 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 $x^{2}+y^{2}=1$ (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	${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$	$(x,y)=(a\cosh t,b\sinh t)$

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

External links

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