Non-Euclidean Geometry

Abstract

In Chapter 3 we saw that the group G which we have been discussing is formed from the transformation $$ y = \frac{{xT + xx'{v_1} + {v_2}}}{{xu{'_2} + xx'b + d}} $$ (1) (And at the same time we have $$ yy'\left( {\frac{{xu{'_1} + xx'a + c}}{{xu{'_2} + xx'b + d}}} \right).) $$ (2) Observe that the matrix $$ M = \left( {\begin{array}{*{20}{c}} T&{u{'_1}}&{u{'_2}}\\ {{v_1}}&a&b\\ {{v_1}}&c&d \end{array}} \right) $$ (3) Satisfies $$ MJM' = J. $$ (4) Thus in terms of homogeneous coordinates we have $$ (\xi *,\eta _1^*,\eta _2^*) = \rho (\xi ,{\eta _1},{\eta _2})M, $$ (5) where M is a Transformation leaving invariant $$ \xi \xi ' - {\eta _1}{\eta _2} = 0. $$ Letting n1 = s1 + s2,n2 = −s1 + s2 then gives $$ \xi \xi ' + s_1^2 - s_2^2 = 0, $$ and dividing this by s2 we obtain an (n + 1)-dimensional unit sphere. Therefore the study of the n-dimensional space expanded through the group e is equivalent to the study of the spherical geometry of the unit sphere in (n + 1)-dimensional space. We shall discuss this type of geometry again when we study mixed partial differential equations later on. However, it may be mentioned that this is just a generalization of the method of stereographic projection which produces a correspondence between the complex plane and the unit sphere.