mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

in a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

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 of ...

, an

affine plane In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are *Euclidean planes, which are affine planes over the real number, reals equipped with a metric (mathematics), metric, the Euclidean dista ...

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 do ...

. 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 curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

s. 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 that a ...

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 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 func ...

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 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 system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

s, 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 th ...

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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

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 ma ...

. 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 In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...

. 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). Part ...

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

Algebraic plane curve

is a curve in an

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...

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 mathematics ...

of the defining equation, which is equal, in case of an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

, 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 ar ...

. For example, the circle given by the equation

x^2 + y^2 = 1

has degree 2. The

non-singular In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...

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 specia ...

s, and their

projective completion In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...

are all

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

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 curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...

s and, if they are non-singular,

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s. Those of degree 4 are called

quartic plane curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of ...

Examples

Numerous examples of plane curves are shown in Gallery of curves and listed at

List of curves This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), physics, engineering, economics, medicine, biology, psychology, ecology, etc. Mathematics (Geometry) ...

. 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