Now consider the two lines AL, and AL2 (Fig. 40) with the equations,

a15+b1n+c1 =0,

a25 +b2n+c2=0.

Any line through their point of intersection A has the equation (a1 + λa2)§ + (b1 + λb2) n + (C1 + λc2) = 0

and this line will be one of the tangent lines AR and AQ, if k2 (a1+λa2)2+k2 (b1+b2)2 - (c1 + λc2)2=0,

i.e., if has either of the values

But

is the anharmonic ratio of the four lines AL1, AL2, 22 AR, and AQ. If A lies outside of the fundamental circle, λ1 and 2 are real, and is imaginary. If A lies on the fundamental conic, λ1=λ2, and p=0. If A lies inside the fundamental conic, 1 and 2 are conjugate imaginary, and is real.

i

The Lobachevskian measure of angle between two lines is 2 times the anharmonic ratio of the two given lines and the two tangents to the fundamental circle from the point of intersection of the two given lines.

50. The study of the circle on the Lobachevskian plane by means of its representation on the Euclidean plane leads to interesting results. We obtain the general equation of the

circle by letting (§1, 71) in Eq. (4), sec. 41, be the fixed coordinates of the centre and letting 2-, 72-7 be the variable coordinates of any point on the circle. The equation is then of the form

(§ 16 +nın − k2)2 = c(52 + y2 — k2),

where c is a constant.

(1)

This is the equation of a conic on the Euclidean plane. Coordinates which satisfy this equation and that of the fundamental circle

which is that of the polar of (1, 1). Since the polynominal §15+nın−k2 appears to the second power in Eq. (1) it follows that Eq. (1) is the equation of a conic which is tangent to the fundamental circle at the points where

the latter is cut by the polar of (51, 71).

There are therefore three cases to consider according as (1, 71) lies outside, on, or inside the fundamental circle.

FIG. 41.

(1) When C (1, 71) is inside the fundamental circle, the polar AB of C does not cut the circle in real points and hence the conic (1) lies entirely in the circle (Fig. 41). This corresponds to the ordinary circle on the Lobachevskian plane.

(2) Whence C(1, 71) lies on the fundamental circle, the polar of C is the tangent to the circle at the point (§1, 71). The conic (1) is then tangent to the circle at the same point (Fig. 42). This corresponds on the Lobachevskian plane to the curve approached by a circle as its centre receded to infinity and its radius becomes infinite. This curve is called a limit

curve or horicycle. Its revolution about one of its infinite radii generates the limit-surface mentioned in sec. 25.

(3) When C(1, 71) is outside the fundamental circle, the polar C cuts the fundamental circle in two points A and B (Fig. 43). The conic (1) is therefore tangent to the fundamental conic, and corresponds on the Lobachevskian plane to a real circle with imaginary centre and radius. The straight line AB is a special case of such a circle.

Draw any line CR through C, intersecting AB in Q. This represents on the Lobachevskian plane a line perpendicular to AB (sec. 47). Now in the Lobachevskian measurement

CR and CQ are constant for all positions of Q on AB. Then QR is constant. That is the locus of Q has all its points equidistant from a straight line AB. This curve is sometimes. called the hypo-cycle.

51. We may make, of course, a representation of the Riemannian geometry on the Euclidean plane with coordinates (, ). But in this case the fundamental circle has the equation

and is imaginary. The geometric properties are therefore not visible to the eye.

XI. RELATION BETWEEN PROJECTIVE AND NON-EUCLIDEAN GEOMETRY

52. We have obtained in secs. 48, 49 a special case of the system of measurement first given by Cayley and recognized by Klein as leading to the non-Euclidean geometries. The general principles can now be quickly stated.

Let us take, on a plane for which the Euclidean geometry holds, 1:2:3 as homogeneous point coordinates and assume a fundamental conic with the equation

a11x21+ɑ22x22 +α33x32 +2a12X1X2+2α23X2X3+2α31X3X1=0 or, more compactly,

Σαικτη = 0. (Aik = Aki).

Let the tangential equation of the same conic be

Σήμα, αμ=0, (4= Ακ),

(1)

(2)

i.e., let Eq. (2) be the condition that the straight line A1X1+A2X2+α373 =0 should be tangent to the conic of Eq. (1). For convenience, let us place

Uaa = ΣАikα¿αk, U33=ΣAikßißk, Uaß= Aikα iẞk•

If P1 and P2 are two points on the plane, and Q and R are the points which the line PP2 meets the fundamental conic, and [P1P2QR] is the anharmonic ratio of the four points PIP2QR, then the Cayleyan projective measure of the distance P1P2 is defined by the equation

P1P2 = M log [P1P2QR],

where M is a constant.

2

Similarly, if AL, and AL2 are two lines intersecting at A, and AR and 4Q are the two tangents from A to the fundamental conic, and [L1L2QR] is the anharmonic ratio of these four lines,

then the Cayleyan projective measure of the angle between AL1 and AL2 is given by the equation

The analytic expression of these measures is found as in secs. 48, 49.

If x1x2 x3 are the coordinates of P1, y1:72: y3 the coordinates of P2, and 1, 2 the roots of

By an easy calculation, we may deduce from this

· (3)

(4)

Also if α1x1+a2x2+a3x3=0 is the equation of AL1, ẞ1x1+ B2x2+3x3-0 the equation of AL2, and 1, 2 the roots of