Hyperbolic geometry

The hyperbolic plane In the earlier chapters we have discussed both Euclidean geometry and spherical (non-Euclidean) geometry, and in this last chapter we discuss a second type of non-Euclidean geometry, namely hyperbolic geometry. Gauss introduced the term non-Euclidean geometry to describe a geometry which does not satisfy Euclid's axiom of parallels, namely that if a point P is not on a line L , then there is exactly one line through P that does not meet L . In spherical geometry, the ‘lines’ are the great circles, and in this case any two lines meet. Hyperbolic geometry is a geometry in which there are infinitely many lines through the point P that do not meet the line L , and it was developed independently by Gauss (in Germany), Bolyai (in Hungary) and Lobatschewsky (in Russia) around 1820. We begin by describing the points and lines of hyperbolic geometry without any reference to distance. We shall take the hyperbolic plane to be the upper half-plane H = { x + iy : y > 0} in ℂ. Notice that the real axis ℝ is not part of H . A hyperbolic line (that is, a line in the hyperbolic geometry) is a semicircle in H whose centre lies on ℝ; such semi-circles are orthogonal to ℝ. However, as our concept of circles includes ‘straight lines’ (see Chapter 14), we must also regard those straight lines that are orthogonal to ℝ as hyperbolic lines.