In analytic geometry

Analytic geometry

Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, an asymptote (icon) of a curve

Curve

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

 is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. In some contexts, such as algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, an asymptote is defined as a line which is tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

 to a curve at infinity.

The word asymptote is derived from the Greek

Greek language

Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

  asymptotos which means "not falling together," from ἀ priv.

Abessive case

In linguistics, abessive , caritive and privative are names for a grammatical case expressing the lack or absence of the marked noun...

 + σύν "together" + πτωτ-ός "fallen." The term was introduced by Apollonius of Perga

Apollonius of Perga

Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

 in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.

There are potentially three kinds of asymptotes: horizontal, vertical and oblique asymptotes.