


(kärtē´zhn)
[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 xaxis and the yaxis. 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 xaxis horizontal and the yaxis vertical, so that the xcoordinate, or abscissa, of P is measured along the horizontal perpendicular from P to the yaxis (i.e., parallel to the xaxis) and the ycoordinate, or ordinate, is measured along the vertical perpendicular from P to the xaxis (parallel to the yaxis). In oblique coordinates the axes are not perpendicular; the abscissa of P is measured along a parallel to the xaxis, and the ordinate is measured along a parallel to the yaxis, 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 xcoordinate represents the distance from the yzplane measured along a parallel to the xaxis, the ycoordinate represents the distance from the xzplane measured along a parallel to the yaxis, and the zcoordinate represents the distance from the xyplane measured along a parallel to the zaxis (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.

