structural:summarization
Analytic geometry

In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form a x + b y + c z + d = 0 {\displaystyle ax+by+cz+d=0}, where a, b, c {\displaystyle a,b,c} and d {\displaystyle d} are real numbers and x, y, z {\displaystyle x,y,z} are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values a, b, c {\displaystyle a,b,c} are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in R 2 {\displaystyle \mathbb {R} ^{2}} or as the solution set of two linear equations with values in R 3. {\displaystyle \mathbb {R} ^{3}.}

A conic section is the intersection of a cone with equation x 2 + y 2 = z 2 {\displaystyle x^{2}+y^{2}=z^{2}} and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.

The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name analytic geometry. This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.

Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.