COORDINATES (from 91L. eoordinare, to coordinate, from Lat. co-, together ± ordinare, to arrange. from o•do, order). Magnitudes which serve to determine the position of an element.— point, line. or plane—relative to some fixed figure. For instance, latitude and longitude are arcs (or angles) that define the position of a ship at sea relative to the equator and the prime meridian ; latitude, longitude, and elevation above sea-level serve to determine the position of a balloon.
The method of treating geometry analytically by use of coordinates is clue chiefly to Descartes (1637), although the terms coOrdinates and axes of coordinates were first used by Leibnitz (1694). For the explanation of rectangular coordinates as used in plane geometry, see ANALYTIC GEOM ETRY. As there explained. the axes in the rec tangular system are at right angles to each other, but it is often more convenient to em ploy a system in which the axes form oblique angles. Coordinates referred to such a system of axes are called oblique coordinates. The notation is the same as in the rectangular sys tem, and the lines which determine a point are drawn parallel to the axes; thus the coOrdinates (x, y) of a point form the outer adjacent sides of a parallelogram of which the axes form the inner sides. Another system in common use is that of polar coltrdinates. This involves two magnitudes—the linear distance from a fixed point and the angular distance from a fixed line.
In the figure the position of point P is deter mined by the distance p from 0, and the angle between p and the fixed line OA. 0 is called the pole and p the polar radius or radius vector. If OA passes through the centre C of a circle, the polar equation of the circle is p = 2reosO. That is, the values of p and 0, which satisfy this equation, determine points on the circle. If C is taken as the pole, the equation of the circle is evidently p = r. Rec tangular coOrdinates may he changed to polar co ordinates, and vice versa, by means of the equa tions is the negative reciprocal of the intercept on the X-axis, and v the negative reciprocal of the inter cept on the 17-axis. For if y = 0, x = and if x = 0, y = — 1. The segment a, in the figure, is the intercept on the X-axis, and sponds to y = 0. and b is the intercept on the V-axis and' corresponds to x = 0. Therefore.
a = — - and b = — -, whence I/ = - .-- and a v — If x and y are regarded as constants and v, v as variables in the equation vx 1 = 0, this is the equation of all lines passing through the point (x, y)—that is, of a pencil of which the point (x, p) is the vertex. Hence
this equation is called the line-equation of the given point, and the system of eo6rdinates one point intercept coOrdinates. Two-point or bi punetual coOrdinates determine the position of an element in the plane by reference to two fixed points and a given direction. As in one point coordinates there are two kinds, line co ordinates and point coordinates, so these classes exist in two-point coordinates. Bipunetual line coordinates are the distances of a variable line taken in a constant direction from two fixed points. Bipunetual point coordinates are, each, the negative reciprocal of the distance measured in a given direction from one of two fixed points to the line determined by the variable point and the other fixed point.
Although two magnitudes are sufficient to fix the position of a point in a plane, the introduc tion of a third has the advantage of rendering homogeneous certain equations involved; how So far we have considered lines as loei of points whose coOrdinates satisfy given relations; but it is often more convenient to select magnitudes which determine lines passing through a given point. Thus vx = 0 may he taken as the equation of a straight line in which a ever, the coordinates of a single element are, in general, connected by a non-homogeneous rela tion. Thus, if x, y, z in the figure represent the perpendicular distances of a point from the sides a, b, e of the triangle ABC, they are con nected by the relation ax ± by + cz = k = 2 X area of ABC, or x sin A ± y sin B z, sin C = a constant. (See TRIGONOMETRY.) We may also take x, y, z oblique to a. b. c. each forming the same angle with its corresponding side. Equa tions between the coordinates of two or more points in this system are homogeneous, as nix + py + qz = 0, the equation of a straight line. Such coordinates are called trilinear or homo geneous coordinates. These form a special case of barycentrie eoOrdivates, the first homogene ous coordinates in point of time, introduced by :Mains (q.v.) in his Der barycentriscbe Cakid ( 1S27). Tetrahedral space coordinates belong to the same class. If in the above figure rectangu lar axes are also assumed, and if P„ are the perpendiculars from the origin upon the sides a, b. c, respectively, then BPC y ACPand •