Curve

There are no fewer than three distinct notions of curve throughout mathematics.

In topology, a curve is a one-dimensional continuum (Charatonik and Prajs 2001).

In algebraic geometry, an algebraic curve over a field is the zero locus of some polynomial of two variables which has its coefficients in .

In analytic geometry, a curve is continuous map from a one-dimensional space to an -dimensional space. Loosely speaking, the word "curve" is often used to mean the function graph of a two- or three-dimensional curve. The simplest curves can be represented parametrically in -dimensional space as

			(1)
			(2)
			(3)
			(4)

Other simple curves can be simply defined only implicitly, i.e., in the form

(5)

When discussing curves from the standpoint of analytic geometry, care must be exhibited to maintain the important distinction between the curve itself and its image within its codomain. For example, the curves , , defined, respectively, by

(6)

and

(7)

are unique as curves even though both functions have the interval as their image in . This distinction is especially important due the fact that unique curves may possess drastically different geometric behavior in terms of self-intersection, etc., despite having identical images.

From the perspective of analytic geometry, the term curve is usually preceded by any of a number of terms to designate certain geometric behavior, e.g., closed curve, simple curve, smooth curve, etc.