Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.
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.
René Descartes introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled 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 language French, and its philosophical principles, provided the foundation for the calculus, that was later introduced by Isaac Newton and Gottfried Wilhelm Leibniz, independently of each other.
Important themes of analytical geometry are:
Many of these problems involve linear algebra.
Analytic geometry, for algebraic geometers, is also the name for the theory of (real or) complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables (or sometimes real ones). It is closely linked to algebraic geometry, especially through the work of Serre in GAGA. It is strictly a larger area than algebraic geometry, but studied by similar methods.