two proofs

 a theorem on saccheri quadrialterals Suppose that ABDC is a saccheri quadrialteral in the hyperbolic plane, having right angles at C and D, and with d(A,C) = d(B,D) = d, d(C,D) = c, d(A,B) = a. Let h be the length of the altitude of ABDC. Then (1) sinh(½a) = cosh(d)sinh(½c), (2) cosh(h)cosh(½a) = cosh(d)cosh(½c). proof We may as well move the picture so C,D lie on the real axis, as shown. We have indicated the hypercircle XY on which A,B lie. (1) Let x = d(B,C), and let θ be the hyperbolic angle BCD. Then as ACD is a right angle, cos(ACB) = sin(θ). We use various results on hyperbolic triangles. By the Cosine Rule for ΔABC, cosh(a) = cosh(x)cosh(d)-sinh(x)sinh(d)sin(θ). By the Sine Formula for ΔBCD, sin(θ) = sinh(d)/sinh(x). By Pythagoras Theorem for ΔBCD, cosh(x) = cosh(c)cosh(d). Combining these, cosh(a) = cosh(c)cosh2(d)-sinh2(d). Using the formulae cosh(t) = 2sinh2(½t)+1, cosh2(d)-sinh2(d) = 1, we get sinh2(½a) = cosh(2(d)sinh2(½c). The result follows. Let M, N be the mid-points of AB, CD, so MN is the altitude. Let NB have hyperbolic length y. By Pythagoras's Theorem applied to NDB, NMB, we get cosh(y) = cosh(d)cosh(½c) = cosh(h)cosh(½a). The result follows. arc length on a hypercircle Suppose that A,B lie on L, a hypercircle of width d associated with the hyperbolic line H. Let C,D be the feet of the perpendiculars from A,B to H, and let a=d(A,B), c=d(C,D). Then L, the hyperbolic length of the arc AB of L, satisfies (1) L = cosh(d)c, and (2) sinh(½a) = cosh(d)sinh(L/2cosh(d)). proof We may as well use the above picture. Suppose we divide CD into N equal hyperbolic segments. Each has length c/N. By the above result, the chord above each segment has length l, where sinh(½l) = cosh(d)sinh(c/2N). There are N chords, so the sum of their lengths is L*= Nl. Thus sinh(L*/2N) = cosh(d)sinh(c/2N). Multiplying through by 2N, we get 2Nsinh(L*/2N) = cosh(d).2Nsinh(c/2N). As N tends to infinity, L* tends to arc length, L and we get L= cosh(d)c. Finally, from the previous result, sinh(½a) = cosh(d)sinh(½c). Substituting for c, we get (2).