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 é 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.