CE5

Cartesian coordinates

Cartesian coordinates [key] [for René Descartes], system for representing the relative positions of points in a plane or in space. In a plane, the point P is specified by the pair of numbers (x,y) representing the distances of the point from two intersecting straight lines, referred to as the x-axis and the y-axis. The point of intersection of these axes, which are called the coordinate axes, is known as the origin. In rectangular coordinates, the type most often used, the axes are taken to be perpendicular, with the x-axis horizontal and the y-axis vertical, so that the x-coordinate, or abscissa, of P is measured along the horizontal perpendicular from P to the y-axis (i.e., parallel to the x-axis) and the y-coordinate, or ordinate, is measured along the vertical perpendicular from P to the x-axis (parallel to the y-axis). In oblique coordinates the axes are not perpendicular; the abscissa of P is measured along a parallel to the x-axis, and the ordinate is measured along a parallel to the y-axis, but neither of these parallels is perpendicular to the other coordinate axis as in rectangular coordinates. Similarly, a point in space may be specified by the triple of numbers (x,y,z) representing the distances from three planes determined by three intersecting straight lines not all in the same plane; i.e., the x-coordinate represents the distance from the yz-plane measured along a parallel to the x-axis, the y-coordinate represents the distance from the xz-plane measured along a parallel to the y-axis, and the z-coordinate represents the distance from the xy-plane measured along a parallel to the z-axis (the axes are usually taken to be mutually perpendicular). Analogous systems may be defined for describing points in abstract spaces of four or more dimensions. Many of the curves studied in classical geometry can be described as the set of points (x,y) that satisfy some equation f(x,y)=0. In this way certain questions in geometry can be transformed into questions about numbers and resolved by means of analytic geometry.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2024, Columbia University Press. All rights reserved.

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