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).
