analytic geometry

Analytic geometry is the of geometry that uses mathematical analysis and calculations for investigating geometric problems. Many such problems can be put into the form of equations, and, by analyzing these equations, one may obtain solutions which can be interpreted geometrically.

The fundamental idea behind analytic geometry is that the position of any point on a plane can be given by an ordered pair of real numbers and any point in space by an ordered triple of real numbers; for this purpose, one has to have a coordinate system which determines the values of the numbers which serve as the coordinates of the points.One of the coordinate system often used in mathematics and physics is the Cartesian system which employs three orthogonal axes. The correspondence of the points on a plane and the ordered pairs (and similarly the points in space and the ordered triples) is a bijection. A locus condition for a line or a curve on the plane as well as for a line, a curve, or a surface in space may then be expressed as an equation or a system of equations.

For example, if we want to study the line which passes through the points corresponding to theordered pairs (http://planetmath.org/CartesianProduct)  $(5,\mathrm{\hspace{0.17em}0})$  and  $(0,\mathrm{\hspace{0.17em}8})$,  one can infer that all points  $(x,y)$  of this line satisfy the equation

$$\frac{x}{5}+\frac{y}{8}=1.$$

The Cartesian coordinate system was introduced and applied by the French mathematician and philosopher René Descartes in 1637 in his work Géométrie. Thus, Descartes is considered to be the founder of analytic geometry.

Title	analytic geometry
Canonical name	AnalyticGeometry
Date of creation	2013-03-22 17:36:23
Last modified on	2013-03-22 17:36:23
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Topic
Classification	msc 51N20
Classification	msc 01A45
Related topic	CartesianCoordinates
Related topic	EuclideanGeometryOfThePlane
Related topic	EuclideanGeometryOfTheSpace
Related topic	LineInThePlane
Related topic	LineInSpace
Related topic	EquationOfPlane
Related topic	Ellipse
Related topic	Hyperbola
Related topic	Parabola
Related topic	QuadraticSurfaces
Related topic	CissoidOfDiocles
Related topic	GeneratricesOfOneSheetedHyperboloid
Related topic	ConicSection