represent parallel hyperbolic lines. The y-axis crosses both semicircles, making a right angle with the unit semicircle and a variable angle Φ with Q. The angle at the center of Q subtended by the radius to (0, y) is also Φ because the two angles have sides that are perpendicular, left side to left side, and right side to right side. The semicircle Q has its center at (x, 0), x < 0, so its radius is 1 − x. Thus, the radius squared of Q is hence The metric of the Poincaré half-plane model of hyperbolic geometry parametrizes distance on the ray {(0, y) : y > 0 } with logarithmic measure. Let log y = a, so y = ea where e is the base of the natural logarithm. Then the relation between φ and a can be deduced from the triangle {(x, 0), (0, 0), (0, y)}, for example: References Marvin J. Greenberg (1974) Euclidean and Non-Euclidean Geometries, pp. 211–3, W.H. Freeman & Company. Robin Hartshorne (1997) Companion to Euclid pp. 319, 325, American Mathematical Society, . Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press, Oxford (See pages 113 to 118). Béla Kerékjártó (1966) Les Fondements de la Géométry, Tome Deux, §97.6 Angle de parallélisme de la géométry hyperbolique, pp. 411,2, Akademiai Kiado, Budapest. Hyperbolic geometry Functions and mappings Angle