Analytic geometry

Ancient Greece

The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry.

Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.

Persia

The 11th century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with Descartes. Omar Khayyamis credited with identifying the foundations of algebraic geometry and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe.

Western Europe

Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes.

Descartes made significant progress with the methods in an essay titled La Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native French tongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.

Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve which satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves.As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.

Coordinates

Main article: Coordinate systems

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:

Cartesian coordinates (in a plane or space)

Main article: Cartesian coordinate system

The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).

Polar coordinates (in a plane)

Main article: Polar coordinates

In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ from the polar axis.

Cylindrical coordinates (in a space)

Main article: Cylindrical coordinates

In cylindrical coordinates, every point of space is represented by its height z, its radius r from the z-axis and the angle θ its projection on the xy-plane makes with respect to the horizontal axis.

Spherical coordinates (in a space)

Main article: Spherical coordinates

In spherical coordinates, every point in space is represented by its distance ρ from the origin, the angle θ its projection on the xy-plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z-axis. The names of the angles are often reversed in physics.

Equations and curves

Main articles: Solution set and Locus (mathematics)

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = xcorresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.

Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x² + y² = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation x² + y² = r² is the equation for any circle centered at the origin (0, 0) with a radius of r.

Lines and planes

Main articles: Line (geometry) and Plane (geometry)

Lines in a Cartesian plane or, more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form:

${\displaystyle y=mx+b\,}$

where:

m is the slope or gradient of the line.

b is the y-intercept of the line.

x is the independent variable of the function y = f(x).