Curve

The term curve has several distinct definitions in mathematics.

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

In algebraic geometry, an algebraic curve over a field is a one-dimensional algebraic variety. An affine plane curve can be defined by a nonconstant polynomial with coefficients in .

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

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

Other simple curves can be defined by an implicit equation of the form

(5)

In analytic geometry, it is important to distinguish the curve itself from its image within its codomain. For example, the curves , , 2, defined by

(6)

and

(7)

are distinct as curves even though both functions have the interval as their image in . Distinct curves may have different geometric behavior, including self-intersection, despite having identical images.

In analytic geometry, qualifiers indicate particular geometric behavior, as in closed curve, simple curve, and smooth curve.