In mathematics, a curve consists of the points through which a continuously moving point passes. This notion captures the intuitive idea of a geometrical one-dimensional object, which furthermore is connected in the sense of having no discontinuities or gaps. Simple examples include the sine wave as the basic curve underlying simple harmonic motion, and the parabola. Curves that close in on themselves, such as a circle, are commonly referred to as a loop. 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 $\backslash mathbb\{R\}$); then a curve $\backslash !\backslash ,\backslash gamma$ is a continuous mapping $\backslash ,\backslash !\backslash gamma\; :\; I\; \backslash rightarrow\; X$, where $X$ is a topological space. The curve $\backslash !\backslash ,\backslash gamma$ is said to be simple if it is injective, i.e. if for all $x$, $y$ in $I$, we have $\backslash ,\backslash !\backslash gamma(x)\; =\; \backslash gamma(y)\; \backslash implies\; x\; =\; y$. If $I$ is a closed bounded interval $\backslash ,\backslash !b$, we also allow the possibility $\backslash ,\backslash !\backslash gamma(a)\; =\; \backslash gamma(b)$ (this convention makes it possible to talk about closed simple curve).

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

A curve $\backslash !\backslash ,\backslash gamma$ is said to be closed or a loop if $\backslash ,\backslash !I\; =\; [a,\; b]$ and if $\backslash !\backslash ,\backslash gamma(a)\; =\; \backslash gamma(b)$. 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. The Jordan curve theorem states that such curves divide the plane into an "interior" and an "exterior".

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 anchor: 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.