Curve

A (simple) curve is characterized by being one-dimensional, continuous, and not self-intersecting. More colloqially, one means by curves only those which are not straight lines.

In mathematics, a curve is typically defined as a subset of an n-dimensional euclidean space which can be continuously mapped bijectively onto the 1-dimensional space of real numbers, R; alternatively, a curve is a subset of Rⁿ which is homeomorphic to R¹.

This definition of curve captures our intuitive notion of a curve as a connected, smooth figure that is "like" a line; although it also includes figures that are not called curves in common usage.

For example, a mathematical curve can have any number of "kinks" or corners, since these have no effect on whether or not the resulting figure can be "stretched" and smoothed into a straight line; this is roughly the meaning of "can be continuously mapped". Some extreme examples of "kinky" curves are the fractal Koch curve and the dragon curve[?].

Curves can be open (for example, a parabola) or closed (like a square). Plane curves[?] are curves that are found in R², curves in even higher dimension are often called space curves[?]. Space curves can form knots and other elaborate structures.

More generally, curves can be defined over any algebraic field. For example, one often considers elliptic curves as the set of points (x, y) satisifying some polynomial in two variables, where both x and y members of some (possibly finite) field K.

Complex Curves