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This editable Main Article is under development and subject to a disclaimer.

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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[edit]

CC Image
Fig. 1. Two-dimensional Cartesian coordinates of point P. The signs of the coordinates in three of the four quadrants are indicated.

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[edit]

CC Image
Fig. 2. Three-dimensional Cartesian coordinates of point P. The frame is right-handed.

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[edit]

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).