ANALYTIC GEOMETRY - OTHER TYPES OF A co-ordinate need not be a distance. In one dimension, a point P of a line can also be specified, for example, by choosing two points of reference A, B, and determining P by the co ordinate X = A P/PB, which is a ratio of lengths, not a length it self. Points within the segment AB have positive co-ordinates, I being that of the midpoint, the points beyond A and beyond B have negative co-ordinates of absolute values < z and > i respectively, and the co-ordinates of A, B and the point at in finity are o, oo and — i . Again, we can choose three points of reference A, B, C, and define P by the value of the cross ratio which is of advantage in dealing with projections.
In plane problems not concerned with actual measurements, the most useful co-ordinates are areal. Instead of two independent co-ordinates, each point has three, connected by a simple iden tity. We refer to a fixed triangle ABC, and the co-ordinates of P are ratios of areas of triangles and are independent of the units of length and area: with the sign convention that x is positive or negative according as A and P are on the same side or opposite sides of BC. Wherever P lies, we have identically The three co-ordinates are equivalent to two independent quan tities only, which we could take to be x and y, but it is more convenient to leave them an unspecified pair of the ratios x : y : z, for the following reason. Since we may write x+y+z for i, we can alter the dimension, in x, y, z jointly, of any term without altering its value. We thus arrange that every equation is homogeneous in the three variables; then it alters none of the work if we replace x, y, z by three other quantities having the same ratios, say, ax, ay, az, and these can be taken to be the co ordinates of P, instead of x, y, z. The sum of these new co-ordi nates is a instead of i, but this does not matter, since we need not use the identity again; the two independent ratios are the same as before. To this extent, the areal co-ordinates of a point are not unique, but each set of co-ordinates always determines a unique point.
The expression for distance in this co-ordinate system is com plicated, involving the sides and angles of the triangle of reference, but it is still true that the equation of a line is of the first degree and conversely. If this is to hold without exception, the particular linear equation x+y+z = o, which contradicts the fundamental identity for the co-ordinates of a finite point, must be intrepreted as a line altogether at infinity, the locus of the points at infinity on each of the other lines of the plane. There are other systems of homogeneous co-ordinates, in which also linear equations represent lines, the line at infinity having a less simple form; for example, lrilincar co-ordinates, which are the perpendicular dis tances of P from the sides of the triangle of reference.
The Cartesian co-ordinates (a, b) of a point P can be regarded as specifying the two lines x = a and y = b, intersecting at P, selected from the two families of lines parallel to the axes 0 Y, OX respectively. We have a more general type of co-ordinate, re placing these two families of lines by other curves; e.g., two families of concentric circles, with fixed centres A1, A2 and vary ing radii. Then if P is given, it determines the values P2) of the radii for which the two circles, one from each family, inter sect at- P. These are bipolar co-ordinates. If are given, P is not uniquely determined, for the two circles through P have another real intersection, the reflection of P in the line In some connections, for example, if we are interested in one half only of the plane, this is no bar to their use.
In each of these systems, all the co-ordinates are elements of the same kind. This is not essential; of the two polar co-ordinates, one is a length and one an angle. In other systems, the inter pretation of a co-ordinate by itself is less simple than that of some combination. For example, one of the circles, which deter mine P in the bipolar system, has a cartesian equation of the form 2ax+ 2by = X, where a, b are constants depending on the fixed centre, and X is a parameter which can serve instead of as a co-ordinate of P, since it determines the circle equally well. Here X itself is the negative square of the tangent from the Cartesian origin to the circle through P; whereas the combina tion sI is the radius