In mathematics, a plane curve is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

in a plane that may be either a

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

, an affine plane or a

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...

. The most frequently studied cases are smooth plane curves (including

piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...

smooth plane curves), and algebraic plane curves. Plane curves also include the

Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition tha ...

s (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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

by an

implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit fun ...

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)

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 In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ...

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 In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to t ...

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 and is a one-dimensional

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...

. This means that a smooth plane curve is a plane curve which "locally looks like a

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

", in the sense that near every point, it may be mapped to a line by a

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...

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

f(x, y) = 0,

where is a

, and the

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...

s and are never both 0 at a point of the curve.

Algebraic plane curve

An algebraic plane curve is a curve in an affine or

given by one polynomial equation

f(x,y) = 0

(or

F(x,y,z) = 0,

where is a

homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...

, in the projective case.) Algebraic curves have been studied extensively since the 18th century. Every algebraic plane curve has a degree, the

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

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 In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are ...

. 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 section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...

s, 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 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 This is a gallery of curves used in mathematics, by Wikipedia page. See also list of curves. Algebraic curves Rational curves Degree 1 File:FuncionLineal01.svg, Line Degree 2 File:Circle-withsegments.svg, Circle File:Ellipse Properties of D ...

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):

References

*. *. *.

External links

* {{Authority control Euclidean geometry es:Curva plana