Plane curve

Template:Short description Template:No footnotes 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 <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 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 Template:Tmath 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 <math>f(x, y) = 0,</math> where Template:Tmath is a smooth function, and the partial derivatives Template:Tmath and Template:Tmath 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 <math>f(x,y) = 0</math> (or <math>F(x,y,z) = 0,</math> where Template:Mvar 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 <math>x^2 + y^2 = 1</math> 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 <math>x^2 + y^2 = 1</math> (that is the projective curve of equation Template:Nowrap 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	<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
Circle	<math>x^2+y^2=r^2</math>	<math>(x,y)=(r \cos t, r \sin t)</math>		framless
Parabola	<math>y-x^2=0</math>	<math>(x,y)=(t,t^2)</math>	<math>y=x^2</math>	File:Parabola.svg
Ellipse	<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1</math>	<math>(x,y)=(a \cos t, b \sin t)</math>		framless
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

See also

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

References

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External links

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