What is a Curve ?

In general topology, when non-differentiable functions are considered, it is the map {\displaystyle \gamma } \gamma , which is called a curve, because its image may look very differently from what is commonly called a curve

For example, the image of the Peano curve completely fills the square. On the other hand, when one considers curves defined by a differentiable function (or, at least, a piecewise differentiable function), this is commonly the image of the function which is called a curve.

A plane curve is a curve for which {\displaystyle X} 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 {\displaystyle X} X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane.

These definitions of plane, space and skew curves apply also to real algebraic curves, although the above definition of a curve does not apply (a real algebraic curve may be disconnected).

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.