Transcendental curve

Inanalytical geometry,atranscendental curveis acurvethat is not analgebraic curve.Here for a curve,C,what matters is the point set (typically in theplane) underlyingC,not a given parametrisation. For example, theunit circleis an algebraic curve (pedantically, the real points of such a curve); the usual parametrisation bytrigonometric functionsmay involve thosetranscendental functions,but certainly the unit circle is defined by a polynomial equation. (The same remark applies toelliptic curvesandelliptic functions;and in fact to curves ofgenus> 1 andautomorphic functions.)

The properties of algebraic curves, such asBézout's theorem,give rise to criteria for showing curves actually are transcendental. For example, an algebraic curveCeither meets a given lineLin a finite number of points, or possibly contains all ofL.Thus a curve intersecting any line in an infinite number of points, while not containing it, must be transcendental. This applies not just tosinusoidalcurves, therefore; but to large classes of curves showing oscillations.

The term is originally attributed toLeibniz.

Further examples

• Cycloid
• Trigonometric functions
• Logarithmicandexponentialfunctions
• Archimedes' spiral
• Logarithmic spiral
• Catenary
• Tricomplex cosexponential