Transcendental curve

Transcendental curve

In mathematics, 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.


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