Plane Geometry - Bruce E. Shapiro

SECTION 51. THE POINCARE DISK MODEL 299Figure 51.2: The Poincare Disk represents the plane as a unit circle centeredat the origin of the normal Euclidean plane. Lines are either diameters orarcs of circles that are orthogonal to the edge of the disk.The above matrix inverse holds so long as ∆ ≠ 0. Thereforex 0 = (1 + r2 1)y 2 − (1 + r 2 2)y 12∆y 0 = (1 + r2 2)x 1 − (1 + r 2 1)x 22∆(51.6)(51.7)Theorem 51.1 Let P = (x 1 , y 1 ) and Q = (x 2 , y 2 ) be the Euclidean coordinatesof two points in the Poincare Disk. Then the unique line connectingthese two points is an arc of the circle with center at (x 0 , y 0 ) (from equations51.6 and 51.7), and radius x 2 0 + y 2 0 − 1.We can define a distance measure as follows.Definition 51.2 Let A and B be any two points in the Poincare Disk, andlet P and Q be the endpoints of the line that connects them. Then theCross-Ratio is defined as[AB, P Q] =(AP )(BQ)(BP )(AQ)where the distances AP , BQ, BP , and AQ are the ordinary Euclideansegment lengths.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.