Asymptote is a term originating from a Greek word that refers to something that has no 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 finding 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 corresponding to the ordinate), 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 of the function with respect to the asymptote line by calculating the cut points of the two. These points will indicate the changes in the position of the function against the asymptote. It is worth mentioning that, although the asymptote and the function are usually represented together, the former is not an integral part of the analytical expression of the latter; for this reason, it is often indicated by means of a dotted line, or it is excluded from the graph.
The utility of asymptotes is found, 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 television tower in Canton, about six hundred meters high), the asymptotic lines confer stability as they function as supports.
Etymology of the term asymptote
The Greek word from which we obtained “asymptote” can be written asymptotos and translated as that does not fall together or simply that does not fall. With respect to its structure, the following parts are distinguished:
* the prefix a-, which can also be found in its an- form. It has a proprietary value that is associated with the meaning of the word “no”, and is appreciated in terms such as anacolut, 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, maladjusted and unheard of, among others;
* the prefix sin-, which can be defined as at once, together or with. We see it, for example, in the words union, synecdoche, phrase and syncretism;
* the root of the Greek verb piptein, the translation of which 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, panache, ask, competition, corduroy, flag, repetition and centripetal, among others;
* the verbal suffix -tos, which refers to a thing that was done or can be done. Some of the terms in which it is found are asbestos, asphalt and antidote.
The famous geometer Apolonio de Perge, born approximately in 262 BC in the city that gave him the surname, was the first to take advantage of the term asymptote to refer to the mathematical concept of a line that fails to touch a hyperbola, in his treatise « About the conic sections «. It is worth mentioning that the names of the parabola and ellipse are also owed, as well as the theory of epicycles (which seeks to give an explanation to the apparent variation in the speed of the Moon and the supposed movement of the planets).
