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# Cartesian Equation

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

 (1)

where the left-hand side is some expression of the Cartesian coordinates , ..., . The -tuples of numbers fulfilling the equation are the coordinates of the points of .

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

 (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

 (3)

Often the letters , , are used instead of indexed coordinates , , .

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

 (4) (5)

For example, the system

 (6) (7)

represents the intersection of the coordinate plane (the set of points for which ) with the coordinate plane (the set of points for which ). The result is the set of points , i.e., the above system represents the -axis.

In general, in the three-dimensional Euclidean space, a single linear Cartesian equation represents a plane, whereas an algebraic surface of order is given by a polynomial equation of degree . 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).

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