, a transcendental curve
is a curve
that is not an algebraic curve
. Here for a curve C
what matters is the point set (typically in the plane
) underlying C
, not a given parametrisation. For example the unit circle
is an algebraic curve (pedantically, the real points of such a curve); the usual parametrisation by trigonometric functions
may involve those transcendental functions
, but certainly the unit circle is defined by a polynomial equation. (The same remark applies to elliptic curves
and elliptic functions
; and in fact to curves of genus
> 1 and automorphic functions
The properties of algebraic curves, such as Bézout's theorem, give rise to criteria for showing curves actually are transcendental. For example an algebraic curve C either meets a given line L in a finite number of points, or possibly contains all of L. Thus a curve intersecting any line in an infinite number of points, while not containing it, must be transcendental. This applies not just to sinusoidal curves, therefore; but to large classes of curves showing oscillations.
Other examples of transcendental curves are plots of cycloids, and the exponential and logarithmic functions.
The term is originally attributed to Leibniz.