ANGLES AND ARCS IN THE HYPERBOLIC PLANE H.S.M. Coxeter Dedicated to H.G. Forder on his 90th birthday (received 17 May, 1979) 1. Introduction This essay may be regarded as a sequel to Chapter V of H.G. Forder's charming little book, Geometry [Forder 1950, pp. 80-95], which is summarized in Sections 2 and 3. Section 4 gives a fuller account of Lobachevsky's approach to hyperbolic trigonometry. Section 5 contains a simple proof that the natural unit of measurement is the length of a horocyclic arc such that the tangent at one end is parallel to the diameter at the other end. Section 6 frees Poincare's half-plane model from its dependence on cross-ratio, by introducing a more natural concept: the inversive distance between two disjoint circles. Section 7 shows how an equidistant-curve of altitude a is represented by lines crossing the absolute line at angles ±n(a) . Finally, Section 8 provides an easy deduction of Poincare's formula \dz\/y for the hyperbolic line- element. B Math. Chronicle 9(1980) 17-33. 17 2. Asymptotic triangles Felix Klein once said that "non-Euclidean geometry ... forms one of the few parts of mathematics which ... is talked about in wide circles, so that any teacher may be asked about it at any moment" [Klein 1939, p. 135]. The richest kind of non-Euclidean geometry is the one discovered independently about 1826 by J. Bolyai (1802-1860) and N.I. Lobachevsky (1793-1856). After 1858, when B. Riemann and L. Schlafli had discovered another kind, Klein named the old kind hyperbolic. This geometry shares with Euclid the first 28 propositions and many more. The first departure occurs when we consider a line i and a point B not on it (see Figure 1). In the plane Bl , the pencil of rays going out from B includes some that intersect the infinitely long line I and others that do not. The two special rays that separate those intersecting I from all the rest are said to be parallel to I . In Euclid's geometry they are the two halves of a single line, but in hyperbolic geometry they form an angle. This angle is bisected by the perpendicular BC from B to I . Lobachevsky called either half the angle of parallelism corresponding to the distance a - BC , and denoted it by II (a) . After proving that parallelism is an equivalence relation it becomes natural to describe the two rays parallel to I as BM and BN where M and N are ideal points: the infinitely distant ends of the line I = MN . (It was D. Hilbert who first called them ends.) We speak of asymptotic triangles BCM , BCN , and call BMN a doubly asymptotic triangle. The limiting form of BMN when B recedes from C towards the end L of the ray CB is the trebly asymptotic triangle LMN (Figure 2): a triangle whose three angles are all zero while its three sides are all infinitely long! 18 --------------- 1-------------- C Figure 2 : A trebly asymptotic triangle Gauss, in his famous letter of March 1832 to Bolyai's father, gave an extremely elegant proof that the area of any triangle ABC is proportional to its angular defeat k - A - B - C [see Coxeter 1969, pp. 297-299]. The second of the seven numbered steps in this proof is the statement that a trebly asymptotic triangle has a finite area. Admirers of Lewis Carroll [Dodgson 1890, p. 14] must face the sad fact that he rejected the possibility of hyperbolic geometry because he found it unthinkable that a triangle (or quadrangle) could retain a finite area when its sides were indefinitely lengthened. This gap in Gauss's proof was finally closed in 1905 by Liebmann [1912, p. 54; see also Coxeter 1969, p. 295]. For Liebmann saw how to dissect a finite quadrangle into an infinite sequence of small pieces that can be reassembled to form an asymptotic triangle. 3. The horocycle and the horosphere Lobachevsky announced his newly discovered world in a lecture at Kazan on the 12th of February, 1826, which he published later as a little book, Geometric investigations on the theory of parallel lines. 19 The Dover edition of Bonola's Non-Euclidean Geometry [Bonola 1955] includes, as an Appendix, G.B. Halsted's translation of that little book. There we can see how cleverly Lobachevsky derived hyperbolic trigonometry from spherical trigonometry (which is independent of considerations of parallel lines). Figure 4 : An arc of a horocycle Figure 3 shows an arc e = CC' of a circle of radius b = AC = AC' . When C stays fixed while b tends to infinity, so that the centre A tends to the end N of the ray CA , the limiting form of the circle is an interesting curve called a limiting curve or boundary line or horocycle (Figure 4), whose diameters form a pencil of parallel lines having N as their common end [Forder 1950, p. 85; Coxeter 1969, p. 300]. (The term horocycle has the same root as horizon.) In three-dimensional space, continuous rotation about the diameter NC yields a surface called a horoephere, which is the limiting form of a sphere whose centre recedes to infinity. All the lines through N , diameters of the horosphere, form a bundle of parallel lines, and each plane through any one of these lines cuts the horosphere along a horocycle. Many of Lobachevsky's results are derived from his observation that the intrinsic geometry of the horosphere is Euclidean: the set of horocycles on the horosphere can be represented isometrically by the set of straight lines in the ordinary Euclidean plane. We shall find a modern proof in Section 6 [see also Liebmann 1912, p. 61; Coxeter 1969, p. 304]. 20 B 4. Lobachevsky's asymptotic orthoscheme In 1620, John Napier, Baron of Merchiston, obtained the = 10 formulae which connect (in threes) the five parts of a right-angled spherical traingle ABC (right-angled at C): namely the three sides a , b , o and the acute angles A , B . For instance, cos a = cos a cos b = cot A cot B [Donnay 1945, p. 40]. What Lobachevsky did was to base a special kind of asymptotic tetrahedron BCAN on a hyperbolic triangle ABC by erecting a ray AN perpendicular to the plane ABC and completing the tetrahedron with parallel rays BN and CN so that the three edges BC , CA , AN are mutually perpendicular, BC is perpendicular to the plane ACN , and Z . CBN = n(a), {-ACN = II(Z>) , Z ABN = JI(c) . This tetrahedron BCAN (Figure 5) is called an asymptotic orthoscheme [Coxeter 1969, p. 156]. 21 Lobachevsky observed that the three planes NBC , NCA , NAB , whose lines of intersection are parallel, cut out, from any horosphere with centre N , a horospherical triangle which is isometric to a Euclidean traingle. Since the first two of these three planes are perpendicular, this triangle is right-angled; therefore its two acute angles, which are the dihedral angles along the edges NA and NB of the orthoscheme, are complementary. But the dihedral angle along NA is equal to the angle A of the hyperbolic triangle ABC ; therefore the dihedral angle along NB is - A . It follows that the three planes BCA , BAN , BNd cut out, from a suitable sphere with centre B , a right-angled epherioal triangle with sides B , 11(c) , 11(a) and acute angles hit - A , II(2>) . In this way, Napier's ten spherical formulae (which remain valid in hyperbolic space) yield a corresponding set of ten hyperbolic formulae. For instance, Napier's cos a = cot A cot B yields (4.1) cos 11(a) = tan A cot n(<?>) . At this stage, n(a:) remains an unknown function of x , decreasing from 11(0) = *sir to n(°°) = 0 , but after some rather formidable work Lobachevsky finds (4.2) tan = e~X . At first he allows this e to be any positive number, but then he chooses a natural unit of measurement by identifying e with the base of Naperian logarithms. Many simpler proofs of this famous formula have been discovered since Lobachevsky's time [for instance, Carslaw 1916, p. 109; Coxeter 1969, p. 453, Figure 16.7a; 1978, p. 395]. It obviously implies cos n(x) = tanh x , cot n(x) = sinh x ; 22 hence (4.1) becomes, in modern notation, (4.3) tanh a = tan A sinh b , analogous to Napier's spherical formula tan a = tan A sin b . 5. The unit horocyclic arc With this background, we can pass easily to some further results concerning circles and horocycles. Consider (in the hyperbolic plane) a regular n-gon with centre A and an inscribed circle of radius b = AC , as in Figure 6. A typical vertex B forms, with A and C , a right-angled triangle ABC in which the side a = BC is half a side of the polygon. By (4.3), in the form tanh a = tan(m/«)sinh b , the perimeter of the n-gon is 23 • * tanh ; > • We may now make a tend to zero and n to infinity (while fixing b) and deduce that the circumference of a circle of radius b is 2ir sinh b . Consequently the area of a circle of radius r is tr 2ir J sinh b db = 2n(cosh r - 1) = 4v sinh2 . 'or Restricting attention to what happens inside the angle A , as in Figure 3, we deduce that a circular arc of radius b subtending an angle A has length (5.1) s = A sinh b and that a sector of radius b and angle A has area (5.2) 2A sinh 2 y = e tanh y .