In analytic geometry, a curve is continuousmap 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.