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 ${\displaystyle f(x,y)=0}$ for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as ${\displaystyle y=g(x)}$ or ${\displaystyle x=h(y)}$ 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 ${\displaystyle (x,y)=(x(t),y(t))}$ for specific functions ${\displaystyle x(t)}$ and ${\displaystyle y(t).}$

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

A smooth plane curve is a curve in a real Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ 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 ${\displaystyle f(x,y)=0,}$ where ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }$ is a smooth function, and the partial derivatives ${\displaystyle \partial f/\partial x}$ and ${\displaystyle \partial f/\partial y}$ 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 ${\displaystyle f(x,y)=0}$ (or ${\displaystyle F(x,y,z)=0,}$ 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 ${\displaystyle x^{2}+y^{2}=1}$ 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 ${\displaystyle x^{2}+y^{2}=1}$ (that is the projective curve of equation ${\displaystyle x^{2}+y^{2}-z^{2}=0}$). 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

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

See also

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

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.