Question

Please write neat and explain thank you.

This problem concerns embedding the complex plane C with elements zx iy in the Riemann sphere defined in 3-dimensional space R' with coordinates (X,Y,Z) as the set of points satisfying X2 + Y2+22 = 1, which is known as the unit sphere and denoted by S2,or in the context of stereographic projection of the complex plane into the sphere, often referred to as the extended complex plane and denoted by C. We identify C with the X,Y-plane by equating z-+iywith (x, y,0), noting that we can and do identify x with X and y with Y, whereas we have used Z for the remaining space coordinate axis perpendicular to the X.Y-plane to distinguish it from the complex number z. Recall that the stereographic projection of z into the Riemann sphere, which we shall denote as P:C-c where (ξ,η, ζ) is defined as the first point of intersection with the sphere of the straight line segment from z - (x,y,0)to the north pole N of (I), (0.0.1), which is also identified with (the point at) co. This line segment, which we denote as I,, may be parametrized as 4:X(I):.tx, r(1)sy, Ζ()s(l-1), o isl. Observe that with the above notation, we may identify the extended complex plane with the adjunction of oo to the complex plane, i.e. C= Cu {o). Theproblem is as follows: (a) Derive the formula for (2); namely, find ξ, η and ζ as functions or. (b) Derive the formula for the inverse of (2); namely, find z as a function of 5.7 and 5

Answer 1

We represent a figure which establishes a 1 - 1 correspondence between the points on sphere and points on plane. This sphere s is said to be Riemann sphere and the projection is called as stereographic projection. For this purpose a point N on the sphere is taken and vertex of projection and the equatorial plane as the plane of projection. Then exist a unique point A' of the plane to any point 'A' on the sphere except the point N, conversely a unique point of the sphere to each point on the plane. The equation of sphere s is x^2 + y^2 + z^2 = 1 and on the plane of projection z = 0, Taking N as (0, 0, 1) [vertex of projection] Let The stereographic projection of z into Riemann sphere is denoted P: k rightarrow 4^infinite p(z) = p(x, y, 0) = (sigma, intersection, tau) where (sigma, intersection, tau) is defined as first point of intersection with the sphere of the straight line