Editing Transcendental curve

In [[analytical geometry]], a '''transcendental curve''' is a [[curve]] that is not an [[algebraic curve]].<ref name="newman">Newman, JA, ''The Universal Encyclopedia of Mathematics'', Pan Reference Books, 1976, {{ISBN|0-330-24396-9}}, "Transcendental curves".</ref> Here for a curve, ''C'', what matters is the point set (typically in the [[plane (mathematics)|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 function]]s may involve those [[transcendental function]]s, but certainly the unit circle is defined by a polynomial equation. (The same remark applies to [[elliptic curve]]s and [[elliptic function]]s; and in fact to curves of [[genus (mathematics)|genus]] > 1 and [[automorphic function]]s.)

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.

The term is originally attributed to [[Gottfried Wilhelm Leibniz|Leibniz]].

== Further examples ==
* [[Cycloid]]
* [[Trigonometric function]]s
* [[Logarithm]]ic and [[exponential function|exponential]] functions
* [[Archimedes' spiral]]
* [[Logarithmic spiral]]
* [[Catenary]]
* [[Quadratrix of Hippias]]

==References==
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[[Category:Curves]]

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