In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments. A large number of other curves have been studied in geometry.

This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).

Definitions

In mathematics, a (topological) curve is defined as follows: let I be an interval of real numbers (i.e. a non-empty connected subset of \mathbb{R}); then a curve \!\,\gamma is a continuous mapping \,\!\gamma : I \rightarrow X, where X is a topological space. The curve \!\,\gamma is said to be simple if it is injective, i.e. if for all x, y in I, we have \,\!\gamma(x) = \gamma(y) \implies x = y. If I is a closed bounded interval \,\!b, we also allow the possibility \,\!\gamma(a) = \gamma(b) (this convention makes it possible to talk about closed simple curve).

If \gamma(x)=\gamma(y) for some x\ne y (other than the extremities of I), then \gamma(x) is called a double (or multiple) point of the curve.

A curve \!\,\gamma is said to be closed or a loop if \,\!I = b and if \!\,\gamma(a) = \gamma(b). A closed curve is thus a continuous mapping of the circle S^1; a simple closed curve is also called a Jordan curve or a Jordan arc.

A plane curve is a curve for which X is the Euclidean plane  these are the examples first encountered  or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). However, in the case of algebraic curves it is very common not to restrict the curve to having points only defined over the real numbers.