On the \(n\)-sphere \(S^{n}\) find coordinates corresponding to (i) stereographic projection, (ii) spherical polars.

Stereographic projection is a method of projecting points from the surface of a sphere onto a plane. We take a sphere of radius 1, without loss of generality, centered at the origin. The north pole N of the sphere is the point (0, 0, 1). Any point P on the sphere not equal to N determines a unique line in \(\mathbb{R}^3\), passing through N and P. This line intersects the plane z=0 in exactly one point P', which we call the stereographic projection of P.\nBy writing this explicitly and setting \(x = r \sin \theta \cos \varphi, y = r \sin \theta \sin \varphi, z = r \cos \theta\), where r is radius, \(\theta\) is the inclination angle and \(\varphi\) is the azimuthal angle, we find that \(r=(1+z)/ (1-z), \varphi=\arctan (y/x),(1+z)=r\cos \theta\).

To find the coordinates in spherical polars, we use the standard conversion formulae between Cartesian and spherical polar coordinates.\nFor a point P = (x, y, z) in Cartesian coordinates, the corresponding point in spherical polar coordinates is given by:\n\(r = \sqrt{x^2+y^2+z^2}\)\n\(\theta = \arccos (\frac{z}{\sqrt{x^2+y^2+z^2}})\)\n\(\varphi = \arctan (\frac{y}{x})\)\nwhere r is the radial distance, \(\theta\) is the polar angle, and \(\varphi\) is the azimuthal angle.