Editing Equation (section)

==Geometry==

===Analytic geometry===
[[File:FunLin 04.svg|thumb|The blue and red line is the set of all points (''x'',''y'') such that ''x''+''y''=5 and -''x''+2''y''=4, respectively. Their [[Intersection (Euclidean geometry)|intersection]] point, (2,3), satisfies both equations.]]{{Main|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 <math> ax+by+cz+d=0</math>, where <math>a,b,c</math> and <math>d</math> are real numbers and <math>x,y,z</math> are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values <math>a,b,c</math> 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 <math>\mathbb{R}^2</math> or as the solution set of two linear equations with values in <math>\mathbb{R}^3.</math>

A [[conic section]] is the intersection of a [[cone]] with equation <math>x^2+y^2=z^2</math> 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]].

===Cartesian equations===
[[File:Cartesian-coordinate-system-with-circle.svg|thumb|right|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is {{nowrap|1=(''x'' − ''a'')<sup>2</sup> + (''y'' − ''b'')<sup>2</sup> = ''r''<sup>2</sup>}} where ''a'' and ''b'' are the coordinates of the center {{nowrap|(''a'', ''b'')}} and ''r'' is the radius.]]In [[Cartesian geometry]], equations are used to describe [[geometric figures]]. As the equations that are considered, such as [[implicit equation]]s or [[parametric equation]]s, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of [[algebraic geometry]], an important area of mathematics.

One can use the same principle to specify the position of any point in three-[[dimension]]al [[space (mathematics)|space]] by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).

The invention of Cartesian coordinates in the 17th century by [[René Descartes]] revolutionized mathematics by providing the first systematic link between [[Euclidean geometry]] and [[algebra]]. Using the Cartesian coordinate system, geometric shapes (such as [[curve]]s) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates ''x'' and ''y'' satisfy the equation {{nowrap|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 4}}.

===Parametric equations===
{{main|Parametric equation}}
A [[parametric equation]] for a [[curve]] expresses the [[coordinates]] of the points of the curve as functions of a [[variable (mathematics)|variable]], called a [[parameter]].<ref>Thomas, George B., and Finney, Ross L., ''Calculus and Analytic Geometry'', Addison Wesley Publishing Co., fifth edition, 1979, p. 91.</ref><ref>Weisstein, Eric W. "Parametric Equations." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParametricEquations.html</ref> For example,
:<math>\begin{align}
x&=\cos t\\
y&=\sin t
\end{align}</math>
are parametric equations for the [[unit circle]], where ''t'' is the parameter. Together, these equations are called a parametric representation of the curve.

The notion of ''parametric equation'' has been generalized to [[Surface (topology)|surfaces]], [[manifold]]s and [[algebraic variety|algebraic varieties]] of higher [[dimension of a manifold|dimension]], with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is ''one'' and ''one'' parameter is used, for surfaces dimension ''two'' and ''two'' parameters, etc.).