Appendices

Equation \(X_{_1}^2+\cdots +X_{n+1}^2=r^2.\) defines a hyper-sphere \(\mathbb S_n\subset \mathbb R^{n+1}=(X_{_1},X_{_2},\ldots ,X_{n+1})\) with radius r and centered at the origin \(O=(0,\ldots ,0)\) . Figure 7.1 represents the stereographic projection from the North pole \(N=(0,0, \ldots ,r)\) onto the Cartesian plane \(\mathbb R^n=(x_i)=(x_{_1},...,x_n)\) tangent to the South pole \(S=(0,\ldots ,-r)\) . A generic point \(B=[X_i(B),X_{n+1}(B)]\) di \(\mathbb S_n\) is projected to the point \(C=[x_i(C)]\) of the plane.