rays, and vice versa. Suppose one agrees to measure length on vertical rays by using logarithmic measure: d((x,y),(x,z)) = |log(z/y)|. Then by means of hyperbolic motions one can measure distances between points on semicircles too: first move the points to Z with appropriate shift and dilation, then place them by inversion on the tangent ray where the logarithmic distance is known. For m and n in HP, let b be the perpendicular bisector of the line segment connecting m and n. If b is parallel to the abscissa, then m and n are connected by a vertical ray, otherwise b intersects the abscissa so there is a semicircle centered at this intersection that passes through m and n. The set HP becomes a metric space when equipped with the distance d(m,n) for m,n ∈ HP as found on the vertical ray or semicircle. One calls the vertical rays and semicircles the hyperbolic lines in HP. The geometry of points and hyperbolic lines in HP is an example of a non-Euclidean geometry; nevertheless, the construction of the line and distance concepts for HP relies heavily on the original geometry of Euclid. Disk model motions Consider the disk D = {z ∈ C : z z* < 1 } in the complex plane C. The geometric plane of Lobachevsky can be displayed in D with circular arcs perpendicular to the boundary of D signifying hyperbolic lines. Using the arithmetic and geometry of complex numbers, and Möbius transformations, there is the Poincaré disc model of the hyperbolic plane: Suppose a and b are complex numbers with a a* − b b* = 1. Note that |bz + a*|2 − |az + b*|2 = (aa* − bb*)(1 − |z|2), so that |z| < 1 implies |(az + b*)/(bz + a*)| < 1 . Hence the disk D is an invariant set of the Möbius transformation f(z) = (az + b*)/(bz + a*). Since it also permutes the hyperbolic lines, we see that these transformations are motions of the D model of hyperbolic geometry. A complex matrix with aa* − bb* = 1, which is an element of the special unitary group SU(1,1). References Lars Ahlfors (1967) Hyperbolic Motions, Nagoya Mathematical Journal 29:163–5 via Project Euclid Francis Bonahon (2009) Low-dimensional geometry : from euclidean surfaces to hyperbolic knots, Chapter 2 "The Hyperbolic Plane", pages 11–39, American Mathematical Society: Student Mathematical Library, volume 49 . Victor