# Cartesian coordinates

In plane analytical geometry, **Cartesian coordinates** are two real numbers specifying a point in a Euclidean plane (a 2-dimensional Euclidean point space, an affine space with distance). The coordinates are called after their originator Cartesius (the Latin name of René Descartes), who introduced them in 1637. In 3-dimensional analytical geometry, a point is given by three real numbers, also called Cartesian coordinates.

## Two dimensions

In two dimensions the position of a point *P* in a plane can be specified by its distance from two lines intersecting at right angles, called axes. For instance, in Figure 1 two lines intersect each other at right angles in the point *O*, the *origin*. One axis is the line *O–X*, the other *O–Y*, and any point in the plane can be denoted by two numbers giving its perpendicular distances from *O–X* and from *O–Y*.

A general point *P* can be reached by traveling a distance *x* along a line *O–X*, and then a distance y along a line parallel to *O–Y*. *O–X* is called the *x-axis*, *O–Y* the *y-axis*, and the point *P* is said to have Cartesian coordinates (*x*, *y*). In the coordinate system shown, as is indicated in the diagram, the *x*-coordinate is positive for points to the right of the *y*-axis and negative for points to the left of this axis. The *y*-coordinate is positive for points above the x-axis and negative for points below it. The coordinates of the origin are (0,0).

One can introduce oblique axes and the position of a point may be defined in the same way: by its distance along lines parallel to the *x* and *y* axes. Sometimes these oblique coordinates also called "Cartesian", but more often the name is restricted to coordinates related to orthogonal axes.

## Three dimensions

In Figure 2 three (black) lines, labeled *X*, *Y*, *Z*, are shown in three-dimensional Euclidean space. They intersect in a point *O*, again called the origin. The lines are the *x*-, *y*-, and *z*-axis. As in the two-dimensional case, the axes consist of two half-lines: a positive and a negative part of the axis. The frame is right handed, if we rotate the positive *x*-axis to the positive *y*-axis the rotation direction (by the corkscrew rule) is the direction of the positive *z*-axis. In older literature and some special applications one may find a left-handed Cartesian set of axes, in which the *x*- and the *y*-axis are interchanged (or, equivalently, the *z*-axis points downward).

The Cartesian coordinates of a point in 3-dimensional space are obtained by perpendicular projection. For example, in Figure 2 a point *P* is shown with projections on the positive parts of the axes, that is, all three coordinate of *P* are positive. The plane *ABCP* is perpendicular to the *x*-axis and *B* is the intersection of the *x*-axis with this plane. The length of *O*–*B* is the *x*- coordinate of the point *P*. The plane *CDEP* is perpendicular to the *y*-axis and the length of *O*–*D* is the *y* coordinate of *P*. Finally, *AFEP* is perpendicular to the *z*-axis and the length of *O*–*F* is the *z* coordinate of *P*.

The planes through the origin *O* spanned by two of the axes are called the *x*-*y* plane (contains the surface *BCDO* of the rectangular block), the *x*-*z* plane (contains the surface *BAFO* of the rectangular block), and the *y*-*z* plane (contains *DEFO*), respectively.

## Higher dimensions

Although orthogonal axes are frequently introduced in Euclidean spaces of dimension *n* > 3, it is unusual to refer to these coordinates as "Cartesian", more commonly they are called coordinates with respect to an orthogonal set of axes (briefly orthogonal, or rectangular, coordinates).