Analytic geometry has two different meanings in mathematics. Except for the section Modern analytic geometry, this article treats the classical and elementary meaning, which is a synonym of coordinate geometry. The modern and advanced meaning refers to the geometry of analytic varieties, whose object is sketched in Section Modern analytic geometry, below.
Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is the foundation of most modern fields of geometry, including algebraic geometry, differential geometry, and discrete and computational geometry, and is widely used in physics and engineering.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions of measurement. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. 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.
History
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 than 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.
The eleventh 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 algebrawith his geometric solution of the general cubic equations, but the decisive step came later with Descartes.
Analytic geometry has traditionally been attributed to René Descarteswho made significant progress with the methods when in 1637 in the appendix entitled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search 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 the foundation for Infinitesimal calculus in Europe.
Abraham de Moivre also pioneered the development of analytic geometry. With the assumption of the Cantor–Dedekind axiom, essentially that Euclidean geometry is interpretable in the language of analytic geometry (that is, every theorem of one is a theorem of the other), Alfred Tarski's proof of the decidability of the ordered real field could be seen as a proof that Euclidean geometry is consistent and decidable.
is an engineering discipline which uses the scientific knowledge of the behavior and effects of electrons to develop components, devices, systems, or equipment (as in electron tubes, transistors, integrated circuits, and printed circuit boards) that uses electricity as part of its driving force. Both terms denote a broad engineering field that encompasses many sub fields including those that deal with power, instrumentation engineering, telecommunications, semiconductor circuit design, and many others.
