From Wikipedia, the free encyclopedia.
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).
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 circle S1; 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 can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger then one (see Koch snowflake) and even positive Lebesgue measure (the last exaaple can be obtained by small variation of the Peano curve constuction). The dragon curve is yet another weird example.
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.
If X is a metric space with metric d, then we can define the length of a curve .
If X is a differentiable manifold, then we can define the notion of differentiable curve. If X is a Ck manifold (i.e. a manifold whose charts are k times continuously differentiable), then a Ck differentiable curve in X is a curve c : I --> X which is Ck (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 Ck differentiable curves c : I --> X and d : J --> X are said to be equivalent if there is a bijective Ck map p : J --> I such that the inverse map p-1 : I --> J is also Ck 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 Ck differentiable curves in X. A Ck arc is an equivalence class of Ck curves under the relation of reparametrisation.
Algebraic curves are also defined, in the setting of algebraic geometry, including the theory of elliptic curves that is now widely applied. This notion of curve is based on the algebra of polynomials, and is not the same as the concept given above.
Definitions
Conventions and Terminology
Rectifiable Curves
A rectifiable curve is a curve with finite length.
A parametrization of c is called natural if for any we have
In Euclidean space, every piecewise continuously differentiable curve is rectifiable and its length is given as the integral of its speed.
Differential Geometry
Other curves
