In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a transcendental curve is a curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

 that is not an algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

. Here for a curve C what matters is the point set (typically in the plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

) underlying C, not a given parametrisation. For example, the unit circle

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

 is an algebraic curve (pedantically, the real points of such a curve); the usual parametrisation by trigonometric function

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s may involve those transcendental function

Transcendental function

A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...

s, but certainly the unit circle is defined by a polynomial equation. (The same remark applies to elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

s and elliptic function

Elliptic function

In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s; and in fact to curves of genus

Genus (mathematics)

In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

 > 1 and automorphic function

Automorphic function

In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group....

s.)

The properties of algebraic curves, such as Bézout's theorem

Bézout's theorem

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...

, 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 Leibniz.

Further examples

• Cycloid
  Cycloid
  A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.It is an example of a roulette, a curve generated by a curve rolling on another curve....
  
  
• Trigonometric function
  Trigonometric function
  In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
  
  s
• Logarithm
  Logarithm
  The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
  
  ic and exponential
  Exponential function
  In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
  
   functions
• Archimedes' spiral
• Logarithmic spiral
  Logarithmic spiral
  A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous...
  
  
• Catenary
  Catenary
  In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...
  
  
• Tricomplex cosexponential