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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. more...

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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 $\text{[math]}$ X is a topological space. The curve $\text{[math]}$ x, $y$ in $I$ , we have $\text{[math]}$ I is a closed bounded interval $\text{[math]}$ Î³(x) = Î³(y) for some $\text{[math]}$ I), then $Î³(x)$ is called a double (or: multiple) point of the curve.

A curve $\text{[math]}$ S¹; a simple closed curve is also called a Jordan curve.

A plane curve is a curve for which X is the Euclidean plane â€” these are the examples first encountered â€” 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 yet another weird example.