Curve

In mathematics, a curve is a geometric object that is one-dimensional and continuous. The idea of dimension is therefore present in the theory of curves, in a simplified form. A large number of special curves have been studied in geometry, starting with the circle. 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 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. 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 circle S¹, if injective it 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 can be hardly called curves in common usage. (In particular image of a simple plane curve can have positive Lebesgue measure. Image of a curve can cover a square in the plane (Peano curve). See also Koch snowflake and the Dragon curve)

Conventions and Terminology

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.

Rectifiable Curves

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

${\mbox{Length}}(c)=\sup \left\{\sum _{i=1}^{n}|c(t_{i})c(t_{i-1})|:n\in \mathbb {N} {\mbox{ and }}a=t_{0}<t_{1}<\dots <t_{n}=b\right\}$

A rectifiable curve is a curve with finite length. In Euclidean space piecewise continuously differentiable curve is rectifiable and its length is given as the integral of its speed.

Differential Geometry

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.

Other Curves

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.