Table of contents |

2 Conventions and Terminology 3 Simple Curves 4 Rectifiable Curves 5 Differential Geometry 6 Other Curves |

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 **R**). Then a curve *c* is a continuous mapping *c* : I --> X, where X is a topological space. The curve *c* is said to be *simple* if it is injective, i.e. if for all *x*,*y* in I, we have *c*(*x*) = *c*(*y*) => *x*= *y*. If I is a closed bounded interval [*a*,*b*], we also allow the possibility *c*(*a*) = *c*(*b*). A curve *c* is said to be *closed* if I = [*a*,*b*] and if *c*(*a*) = *c*(*b*). A closed curve is thus a continuous mapping of the unit circle *S*^{1}. A simple closed curve is also called a Jordan curve.

This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that are not called curves in common usage.

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading.

Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

It can be shown that a topological space X is the image of a simple curve if and only if X is a connected T1 space with at least two elements, satisfying the following property:

- If T is a subset of X which is closed and whose boundary consists of three distinct elements such that the boundary of the interior of T consists of these three elements as well, then T is the union of two disjoint non-empty closed subsets of X.

If X is a metric space with metric d, then we can define the *length* of a curve *c* in X.

If X is a differentiable manifold, then we can define the notion of *differentiable curve*. If X is a *C*^{k} manifold (i.e. a manifold whose charts are *k* times continuously differentiable), then a *C*^{k} differentiable curve in X is a curve *c* : I --> X which is *C*^{k} (i.e. *k* times continuously differentiable). If X is a smooth manifold (i.e *k* = ∞, charts are infinitely differentiable), and *c* is a smooth map, then *c* is called a *smooth curve*. If X is an analytic manifold (i.e. *k* = ω, charts are expressible as power series), and *c* is an analytic map, then *c* is called an *analytic curve*.

A differentiable curve is said to be *regular* if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two *C*^{k} differentiable curves *c* : I --> X and *d* : J --> X are said to be *equivalent* if there is a bijective *C*^{k} map *p* : J --> I such that the inverse map *p*^{-1} : I --> J is also *C*^{k} and *d*(*t*) = *c*(*p*(*t*)) for all *t*. The map *d* is called a *reparametrisation* of *c*, and this makes an equivalence relation on the set of all *C*^{k} differentiable curves in X. A *C*^{k} *arc* is an equivalence class of *C*^{k} curves under the relation of reparametrisation.

A fractal curve is a topological curve with fractional dimension. Since there are different definitions of fractal dimension, there are different definitions of fractal curve. Popular examples of fractal curves include the Koch snowflake and the Dragon curve.

Curves are also defined in the setting of algebraic geometry and the theory of elliptic curves. This notion of curve is algebraic and not the same as the concept given above.