Plane Geometry - Bruce E. Shapiro

Section 51The Poincare Disk ModelIn the Poincare Disk Model of hyperbolic geometry the plane is representedby the unit disk (the disk of radius 1 centered at the origin). Linesare diameters of the unit circle or the portions of arcs of circles that areorthogonal to the unit disk that fall within the unit disk.Two intersecting circles are said to be orthogonal if their tangents meetat 90 ◦ angles (figure 51.1).We illustrate this in figure 51.2. In this illustration, lines l, m and n areall parallel to k. Lines l and m both intersect at the point A, and lines mand n intersect at point B, given two examples of multiple parallel lines tothe same line passing through a single point. The figure also illustrates thefailure of the transitivity of parallelism; although l and m are parallel to k,they are not parallel to each other; furthermore, while l is parallel to n, mis not parallel to n.Angle measure is given by the Euclidean angle measure of the angleformed by the tangent lines at the point of intersection (see figure 51.3).To prove that there is a unique line (given by the circular arc) through anytwo points, we will use analytic geometry to find the general equation fora circle that is orthogonal to the unit circle at the origin. Let the circlehave radius a and center (x 0 , y 0 ). Then from analytic geometry we havethe equation of the circle is(x − x 0 ) 2 + (y − y 0 ) 2 = a 2 (51.1)From figure 51.1 we see by an application of the Pythagorean theorem thatx 2 0 + y 2 0 = a 2 + 1 (51.2)297