The notion of asymptote is used in the field of geometry.
Asymptote is a term originating from a Greek word that refers to something that does not have a coincidence . The concept is used in the field of geometry to name a line that, as it extends indefinitely, tends to approach a certain curve or function , although without being able to find it.
This means that, as the line and the curve extend, the distance between them will tend towards zero . According to their characteristics, asymptotes can be classified as horizontal (when the line is perpendicular to the axis that corresponds to the ordinates), vertical (the line, in this case, is perpendicular to the axis corresponding to the abscissa) or oblique (they are not perpendicular or parallel to any axis).
It is possible to determine the relative position that the function occupies with respect to the asymptote line by calculating the intersection points of the two. These points will indicate the modifications in the position of the function compared to the asymptote. It is worth mentioning that, although the asymptote and the function are usually represented together, the first is not an integral part of the analytical expression of the second; For this reason, it is often indicated by a dotted line or excluded from the graph.
Etymology of the term asymptote
The Greek word from which we obtained "asymptote" can be written asymptotos and translated as that which does not fall together or, simply, that which does not fall together . With respect to its structure , the following parts are distinguished:
* the prefix a- , which can also be found in its form an- . It has a private value that is associated with the meaning of the word "no", and is appreciated in terms such as anacoluto , anarchy , apathetic and analgesic . When combined with the root ne- , of Indo-European origin, which in turn is found in the prefix in- , which comes from Latin, we obtain incapable , unadapted and unheard of , among others;
* the prefix sin- , which can be defined as at the same time , together or with . We see it, for example, in the words union , synecdoche , syntagm and syncretism ;
* the root of the Greek verb piptein , whose translation is to fall . This is linked to the root pet- (of Indo-European origin and with the meanings fly or fall ), which we find in the terms with Latin roots peña , panaché , ask , competition , corduroy , banner , repetition and centripetal , among others;
* the verbal suffix -tos , which refers to a thing that was done or that can be carried out. Some of the terms in which it is found are asbestos , asphalt and antidote .
An asymptote line is one that, when extended indefinitely, tends to approach a function or curve, although without reaching it.
The contributions of Apollonius of Perge
The famous geometer Apollonius of Perge, born approximately in the year 262 BC in the city that gave him his last name, was the first to use the term asymptote to refer to the mathematical concept of a line that fails to touch a hyperbola , in his treatise « On conic sections ».
It is worth mentioning that the names of the parabola and the ellipse are also due to it, as well as the theory of epicycles (which seeks to provide an explanation for the apparent variation in the speed of the Moon and the supposed movement of the planets).
Usefulness of an asymptote
The usefulness of asymptotes is, for example, when representing a curve graphically. These lines, which indicate future behavior and provide support for the curve, can be expressed analytically according to the reference system in question.
This knowledge is usually put into practice in fields such as engineering or architecture . In a hyperboloid structure (such as the famous Canton television tower, about six hundred meters high), the asymptotic lines confer stability since they function as support.