If the actual values are a, B, y, so that a =λx, etc., then 2 a2 + ß2 + k2y2 = k2 = \2 [(n1§ 2 − ~2Š1)2 + (Š1§2 − §2§1)2 2 2 + k2 (§1N2 − §2N1)2] = \ 2k2 [ ( §12 + n12 + §12/k2) (§ 22 +n22 + §22/k2) =\2k2[1 − cos2 ]; therefore λ = cosec &, − ( §1Ê2 + N1N2 +$182/k2)2] where & is the angle between the lines. If the coordinates of the point of intersection satisfy the equation x2+ y2+k2x2=0, i.e. if the lines intersect on the absolute, A is infinite and is zero. The two lines in this case are parailel. If the ratios of the coordinates make x2 + y2 + k2z2<0, A is imaginary. The two lines have then no real point of intersection, and the angle is imaginary. The lines may be said to intersect outside the absolute. (These two cases can, of course, only happen in hyperbolic geometry.) In the latter case the two lines have a common perpendi cular. Let Ex+ny +82-0 be perpendicular to both; then §§1 +nn1 + §§1/k2=0, §Ê2+n2+ §§ 2/k2 = 0 ; therefore §:n:§ = ~182 − ̃1⁄2§1 : §1§2 − §2§1 : k2 (§1~2 – §271) ; but this line is just the polar of their point of intersection. The length p of the common perpendicular is equal to ko, and we have S152 1.2 : includes (1) real actual points, for which the ratios x y z are real and x2 + y2 + k2z2 has the same sign as k2 (2) real ideal points, for which the ratios xyz are real, and (x2+ y2+k22)/k2 is negative, (3) imaginary points, for which at least one of the ratios a Y : z is imaginary. The line joining a pair of conjugate imaginary points is a real line, actual, at infinity or ideal. The distance between a pair of conjugate imaginary points is real only if their join is ideal. 9. Line joining two points. Similarly the line-coordinates of the line joining two points (x1, 1, 1), (≈2, Y2, Z2) are proportional to y1Z2 − Y 2Z1, Z1X2 − Z2X1, X1Y2-X2Y1. The actual values of the lined coordinates are found by multiplying by the factor cosec where d is the distance between the two points. 2 If the ratios of the line-coordinates satisfy the equation 2 Ê2 + 1)2 +¿2/k2 =0, the line is at infinity, and the distance d is zero. If the ratios make §2+n2 + §2/k2<0, the line is wholly ideal, and the distance d is imaginary. 10. Minimal lines. When the join of two points is a tangent to the absolute, the distance between the two points is zero. For this reason the tangents to the absolute are called minimal lines. In euclidean geometry the distance between two points (x1, Y1), (X2, y2) is zero if (X 2 − X1)2 + (Y2 − Y1)2 = 0, i.e. if the join of the two points passes through one of the circular points (Chap. II. § 17). The line at infinity itself passes through both of the circular points, and it is the only real line which passes through them. The distance between two points at infinity should thus be zero. But again, any point on the line at infinity is in Iv. 11] CONCURRENCY AND COLLINEARITY 135 finitely distant from any other point. Hence the distance between two points, both of which are at infinity, becomes indeterminate. In relation to the rest of the plane we must consider such distances as infinite, and the geometry of points at infinity becomes quite unmanageable. The geometry upon the line at infinity by itself, however, is really elliptic, since the absolute upon this line consists of a pair of imaginary points; the "distance" between two points at infinity could then be represented by the angle which they subtend at any finite point. 11. Concurrency and collinearity. The condition that the lines (1, 1, 1), etc., be concurrent is કું, ‰ı E2 na S2 0. ₤3 13 €3 The condition that the points (x1, Yr, Z1), etc., be collinear is These conditions are, of course, the same as those in ordinary analytical geometry, with homogeneous coordinates. Since the equation of a straight line is homogeneous and of the first degree in the coordinates, all theorems of ordinary geometry which do not involve the actual values of the coordinates, or the distance-formulae, will be true also in non-euclidean geometry. These theorems are those of projective geometry. The difference between euclidean and non-euclidean geometry only appears in the form of the identical relation which connects the point and line coordinates, i.e. in the form of the absolute. 12. The circle. A circle is the locus of points equidistant from a fixed point. Let (x, y1, 21) be the centre and r the radius; then the equation of the circle is This equation is of the second degree, and from its form we see that it represents a conic touching the absolute (xx)=0) at the points where it is cut by the line (xx)=0. (xx1)=0 is the polar of the centre, and is therefore equidistant from the circle, i.e. it is the axis of the circle. Hence A circle is a conic having double contact with the absolute; its axis is the common chord and its centre is the pole of the common chord. The equidistant-curve. Let (1, 1, 1) be the coordinates of the axis, and d the constant distance; then the equation of the curve is This again represents a conic having double contact with the absolute, the common chord being the axis. The pole of the axis is equidistant from the curve, and so the equidistant-curve is a circle. In elliptic geometry both centre and axis are real, in hyperbolic geometry the centre alone is real for a proper circle, and the axis alone is real for an equidistant-curve. IV. 13] THE CIRCLE 137 The horocycle. In hyperbolic geometry, the equation of the absolute being x2+ y2 - k22=0, the equation of a horocycle is of the form 13. Coordinates of a point dividing the join of two points into given parts. If (X1, Y1,71), (X2, y2, Z2) are any two points, the coordinates of any point on the line joining them are (λ + μ2, xy, + μψε, λ2 + μ2), for if ax+by+cz=0 is the equation of the line, so that it is satisfied by the coordinates of the two given points, it will be satisfied also by the coordinates of any point with coordinates of this form. Similarly, if we consider these as the line-coordinates of two lines, the coordinates of any line through their point of intersection are of this form. In fact the line λ(α1x +b1y+c2z) +μ(a2x+b2y+c22) = 0, whose coordinates are (\a1+μɑ2,...), passes through the intersection of the two given lines ax+by+c1z=0 and A 2X + b2y +C2z=0. To find the coordinates of a point dividing the join of two points whose actual coordinates are (x1, Y1, Z1) and (X2, Y2, Z2) into two parts 1 and r2, where r1 +~1⁄2=r. 1 2 γ. Let (\x1 + μx2, ...) be the actual coordinates of the required point. Then X1(\X1 +μx2) +Y1 (^\Y1 +μ¥2) + k2z1 (\21 + μz2) = k2 cos r1 k |