curved adj : not straight; having or marked by a curve or smoothly rounded bend; "the curved tusks of a walrus"; "his curved lips suggested a smile but his eyes were hard" [syn: curving] [ant: straight]
- Having a curve or curves.
having a curve or curves
- French: courbé
- past of curve
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).
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). 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 \,\![a, 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 = [a, 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.
This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, without thickness and drawn without interruption, although it also includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a square in the plane (space-filling curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is another unusual example.
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.
Lengths of curves
If X is a metric space with metric d, then we can define the length of a curve \!\,\gamma : [a, b] \rightarrow X by
- \mbox (\gamma)=\sup \left\
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