Page:The New International Encyclopædia 1st ed. v. 05.djvu/441

* COORDINATES. 381 COORDINATES. COORDINATES (from ML. coordinare, to cociiiliiuiU', from Lat. co-, together + ordinare, to arriinge. from unio, order). llagnitiuK's which btrve to dcterniiue the position of an element — ■ point, line, or plane — relative to some fixed figure. For instance, latitude and longitude are ares (or angles) that define the position of a ship at sea relative to the equator and the prime meridian : latitude, longitude, and elevation aliovc sea-level ser-e to determine the position of a halloon. The method of treating geometry analytically by use of coordinates is due chiefly to Descartes (1G.37), although the terms coordinates and axes of coordinates were first used by Leibnitz ( 1G94). For the explanation of rectangular coiirdinates as used in plane geometry, see Analytic Geom- KTRY. 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- j)loy 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 sj'S- tem, and the lines which determine a point are drawn parallel to the axes; thus the coordinates ^ {x, ij) 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 coordinates. 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 O, and the angle 6 between p and the fl.xed line OA. O is called the pole and p the polar radius or radius vector. If OA passes tlirough the centre C of a circle, the polar equation of the circle is p = 2rcos5. 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 coiirdinates may be changed to polar co- ordinates, and vice versa, by means of the equa- tions X = p cos 8, y P = / x^ +y-, = p sin 0; e = tan- V So far we have considered lines as loci 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 jix + vy + 1=0 may be taken as the equation of a straight line in which u is the negative reciprocal of the intercept on the X-axis, and v the negative reciprocal of the inter- cept on the Y-axis. For if y = 0, a" ^ ■> and if ic = 0, j/ = — The segment a, in the figure, is the intercept on the X-axis, and corre- sponds to 1/ = 0, and b is the intercejit on the Y-axis and' corresponds to a; = 0. Therefore. a = — ~- and 6 = — r, whence u — — i and u V a 1' = — _. If x and y are regarded as constants 6 and «, V as variables in the equation iix + iii/ -|- 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, y) is the vertex. Hence tliis equation is called the line-equation of the given point, and the system of coiirdinates 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 ai'e two kinds, line co- ordinates and point coordinates, so these classes exist in two-point coordinates. Bipunctual line coordinates are the distances of a variable line taken in a constant direction from two fixed points. Bipunctual 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- ever, the coordinates of a single element are, in general, connected by a non-homogeneous rela- tion. Thus, if X. y. ; in the figure represent the perpendicular distances of a point from the