Analytic geometry

Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions.

That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.

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.

[1] 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.

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 (expressed in words) 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.

Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.

[3] The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra[4] with his geometric solution of the general cubic equations,[5] but the decisive step came later with Descartes.

[4] Omar Khayyam is 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 analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe.

[6] Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.

La Geometrie, 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.

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.

It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.

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).

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.

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.

For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

where: In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points

Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation

As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space

In analytic geometry, geometric notions such as distance and angle measure are defined using formulas.

The intersection of these two circles is the collection of points which make both equations true.

Traditional methods for finding intersections include substitution and elimination.

Tangent is the linear approximation of a spherical or other curved or twisted line of a function.

Informally, it is a line through a pair of infinitely close points on the curve.