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Cartesian Equation


An equation representing a locus L in the n-dimensional Euclidean space. It has the form

 L:f(x_1,...,x_n)=0,
(1)

where the left-hand side is some expression of the Cartesian coordinates x_1, ..., x_n. The n-tuples of numbers (x_1...,x_n) fulfilling the equation are the coordinates of the points of L.

CartesianCircleSphere

For example, the locus of all points in the Euclidean plane lying at distance 1 from the origin is the circle that can be represented using the Cartesian equation

 x^2+y^2-1=0.
(2)

Similarly, the locus of all points of the three-dimensional Euclidean space lying at distance 1 from the origin is a sphere of radius 1 centered at the origin can be represented using the Cartesian equation

 x_1^2+x_2^2+x_3^2-1=0.
(3)

Often the letters x, y, z are used instead of indexed coordinates x_1, x_2, x_3.

The intersection of two loci L_1 and L_2 is the set of points whose coordinates fulfil the system of equations

L_1:f_1(x_1,...,x_n)=0
(4)
L_2:f_2(x_1,...,x_n)=0.
(5)

For example, the system

L_1:x_1=0
(6)
L_2:x_2=0
(7)

represents the intersection of the coordinate plane x_2x_3 (the set of points for which x_1=0) with the coordinate plane x_1x_3 (the set of points for which x_2=0). The result is the set of points (0,0,x_3), i.e., the above system represents the x_3-axis.

CartesianEquationIntersections

In general, in the three-dimensional Euclidean space, a single linear Cartesian equation represents a plane, whereas an algebraic surface of order n is given by a polynomial equation of degree n. Curves are represented as the intersection of two surfaces. For example, lines are represented as the intersection of two planes, circles as the intersection of a sphere and a plane (or of two spheres). Of course, a given curve can be realized by intersection in infinitely many ways, which correspond to infinitely many different equivalent systems of equations representing the same curve. In any case two equations are needed since a single Cartesian equation can represent a curve only in the plane.

An alternative way to represent a locus is to use parametric equations. Cartesian equations of lines can be derived from parametric ones by algebraic elimination of the parametric variable(s).


See also

Affine Variety, Cartesian, Cartesian Coordinates, Cartesian Geometry, Coordinate System

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Cartesian Equation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CartesianEquation.html

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