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)

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