Cartesian orthogonal coordinate system

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A rectilinear system of coordinates in a Euclidean space.

On a plane, a Cartesian rectangular coordinate system is defined by two mutually-perpendicular lines, the coordinate axes, on each of which a positive direction has been chosen and a segment of unit length has been specified. The point of intersection ($0$) of the coordinate axes is said to be the coordinate origin. One of the coordinate axes $(0x)$ is said to be the abscissa axis; the other one ($0y$) is said to be the ordinate axis. The coordinate axes subdivide the plane into four equal regions, called quarters or quadrants.

The rectangular Cartesian coordinates of a point $M$ are represented by an ordered pair of numbers $(x,y)$, the first of which (the abscissa) is equal to the magnitude of the orthogonal projection of the directed segment $0M$ on the abscissa axis, the second one (the ordinate) being the orthogonal projection of the directed segment $0M$ on the ordinate axis.

A Cartesian rectangular coordinate system in a three-dimensional space is defined similarly to the case of the plane: by an abscissa axis, by an ordinate axis, by an applicate axis, and by a coordinate origin $0$. The plane passing through the coordinate axes are said to be the coordinate planes. They subdivide the space into eight regions, the octants.

A skew-angled (general) Cartesian coordinate system is also occasionally used. As distinct from the rectangular system, the angles between the coordinate axes need not be straight angles.

Named in this way after R. Descartes [1], who introduced the method of rectilinear coordinates.


[1] R. Descartes, "Geometria" , Leiden (1649)
How to Cite This Entry:
Cartesian orthogonal coordinate system. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article