ANALYTIC GEOMETRY - TRANSFORMATIONS A co-ordinate is always measured with respect to a fixed origin or base or frame of reference of some sort. The analytical method gains by choosing the frame so as to simplify the treatment of the problem in hand, and in particular to take advantage of any symmetry. If there is one fixed point outstanding among the data, we ma); take this as cartesian origin. If there is a pair of equal importance, we may put the origin midway between them, and take the axis of x to join them, so that their co-ordinates are (a, o), (—a, o) . If the problem is descriptive and involves a triangle, areal co-ordinates are indicated, and so on. For ex ample, the property of the existence of the centre of gravity of a triangle, where the medians concur, becomes the mere statement that there exists a point whose areal co-ordinates are (I, i, 1).
We often need to change the frame of reference in the middle of a piece of work, and to express, in terms of the new system, any result already obtained in terms of the old co-ordinates. We must find expressions to substitute for each of the old co-ordi nates separately, that involve the new co-ordinates and the con stants determining the new frame of reference in relation to the old, but that do not involve the other one of the old co-ordinates. Thus the change from cartesian to polar co-ordinates, with the same origin and axis, is effected by x = r cos 0, y = r sin 0; and the change back again by r = / 9 = y/x.
If we change from one rectangular frame of axes XOY to another X'0' Y', we must know the co-ordinates OA = a, AO' = b of the new origin 0' referred to the old axes, and also the angle 4) between the O'X' and OX. Then if P is any point, and (x, y), (x', y') its two pairs of co-ordinates, we find: cos 4)—y' sin 4), sin 4)+y' cos 4); and if, at a later point in the work, we wish to translate results involving x', y' into terms of the old co-ordinates we must use x' = (x — a) cos 4+ (y— b) sin 4), y' = — (x —a) sin 4 + (y — b) cos 4), so as to eliminate the new co-ordinates.
If x', y' are point co-ordinates in a plane, 7r', and we make a substitution of the form x'=f1(x, y), y), (I) and interpret x, y as point co-ordinates in another plane 7r, then to any point P of it there corresponds the particular point P' of ir', homologue of P, whose co-ordinates are given by (I) . If P moves and describes a curve k, then P' also moves and describes a curve k', which corresponds to k, and whose properties can be deduced from those in the other plane. If P' is a given point of ir', there corresponds to it all the group of points whose co-ordinates satisfy (I), when in these equations we regard x', y' as known and x, y as unknown. If P' lies on k', in general only one of the group lies on k. Thus the two curves are transformed into one another so that each point of either corresponds to one and only one of the other. In particular, if (I) are such that they can be solved in the same form x= f y'), y= y'), then there is a I, I correspondence between all the points of the two planes, and not only between the points of the two curves.
The degree of a I, I plane transformation is that of the curve in either plane which corresponds to a line in the other. If the two planes are identical, we have a transformation of the plane into itself. The simplest examples are translation, rotation, similarity and their combinations, which are all linear trans formations, and inversion, which is quadratic. The last trans forms P into the point P' of OP such that OP • OP' =constant; it can be expressed by and the reverse equations have the same form. Two different maps of the same region of the earth's surface are I, I transform ations of it and of each other. For example, one may show the whole world as a rectangle, and the other may show it as a pair of circles. The shapes of Australia are quite different; but each spot within one map corresponds to one and only one within the other. This is not true of points on the boundaries, which are exceptional. (See CONFORMAL REPRESENTATION.) So far we have considered co-ordinates of points only; but any variable geometrical object can have co-ordinates, for ex ample, the lines or circles of the plane. If x, y are the cartesian co-ordinates of a point P, the equation of any line l can be taken in standard form ax+33y = I, where a, $ are the reciprocals of the intercepts which l makes on the axes. For different values of a, 0, we can identify 1 with any given line: thus a, j3 may be taken as the line co-ordinates of 1. If x, y are constants and a, 13 variable, the equation expresses that the variable line 1 passes through the fixed point P, and is the equation of the point in line co-ordinates. Just as any equation in point co-ordinates defines a continuous set of points lying on a locus, which is a line if the equation is of degree I, so an equation in line co-ordi nates defines a continuous set of lines all touching a curve, called an envelope from this point of view, which is a point if the equa tion is of degree I. For this reason, line co-ordinates are often called tangential co-ordinates. In general, the degree of the en velope is the number of tangents, real or imaginary, which can be drawn to the curve from any given point. This number is called the class of the curve, and is equal to the degree n if n = or 2, but for n? 3, the two are not generally the same.
Any piece of analysis, which has a geometrical meaning when a pair of variables are taken to be the co-ordinates of a point, has another meaning when the same variables are taken instead to be the co-ordinates of a line; and all such geometrical theorems occur in pairs, dual to each other. Corresponding to any plane figure built up from points and the lines joining them, there is a dual figure, built up in the same way from lines and their points of intersection. For any statement about the one, there is a statement about the other, exactly the same except that points and lines are replaced by lines and points; and the analytical proofs are the same, except for the definitions of the symbols. The perpendicular distance of the line t from 0 is p where 1 the line equation is satisfied by any line at unit distance from 0 and the envelope represented by this equation is the circle of unit radius centre 0. The same equation in point co-ordinates represents the same circle, which is a self dual figure.