Stereographic Projection, Riemann Sphere and Mapping

A plane is inserted through the equator of a unit sphere.
A point on the sphere is mapped onto the plane by creating a line from the point on the sphere, through the north pole, where this line hits the plane is the projection of the point.
Show that circles on the sphere map to circles on the plane except when the circles runs through the north pole, in this case it maps onto a line.

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A Stereographic Projection

Consider a sphere of radius 1, with a plane inserted through the equator

We map the sphere onto the plane by mapping a given point P onto Q, such that P, Q and the North Pole N all lie on the same line. Given this mapping the southern hemisphere maps inside the circle of radius 1, the equator stays where it is and the north hemisphere maps onto the region outside the unit circle. Note that the North Pole maps onto "infinity".

Let's say the point P has coordinates (x,y,z), Q lies at (a,b,0) and N at ...

Solution Summary

Stereographic Projection, Riemann Sphere and Mapping are investigated. The solution is detailed and well presented. Diagrams are included.

1. Let z and z' be points in C with corresponding points on the unit sphere Z and Z' by stereographic projection. Let N be the north pole N(0,0,1).
a) Show that z and z' are diametrically opposite on the unit sphere iff z(z bar)'=-1
ps. here z bar means conjugate of z
b) Show that the triangles Nz'z and NZZ' are similar. The

I have these problems from Topology of Surfaces by L.Christine Kinsey: the problems I require assistance with are 2.26, 2.28, 2.29, and 2.32. These are stated below.
PROBLEM (Exercise 2.26). Describe what stereographic projection does to
(1) the equator,
(2) a longitudinal line through the north and south poles,
(3) a tr

Prove algebraically that the stereographic projection of a circle (C) lying in a sphere (S) is either a circle or a straight line.
Definition:
Stereographic projection is done from the north pole of the sphere onto a plane tangent to the sphere at its south pole.
Hint:
A circle on the sphere is contained in a plane (P),

Let V be a circle lying in S. Then there is a unique plane P in R^3 such that p /\ S = V ( /\ = intersection). Recall from analytuc geomerty that
P = { (x_1,x_2,x_3) : x_1 b_1 + x_2 b_2 + x_3 b_3 = L, where L is a real number}.
Where ( b_1,b_2,b_3) is a vector orthogonal to P . It can be assumed that (b_1)^2 + (b_2)^2 + (b_

Please see the attached file for the fully formatted problems.
Let V be a vector space of all real continuous function on closed interval [ -1, 1]. Let Wo be a set of all odd functions in V and let We be a set of all even functions in V.
(i) Show that Wo and We are subspaces and then show that V=Wo⊕We.
(ii) Find a pro

Please see the attached file for the fully formatted problems.
1. Let P1 and P2 be two points on the unit sphere x2 + y2 + z2 = 1, and w and w2 the corresponding points on the plane
z = 0 under stereographic projection. Show that if P1 and P2 are antipodal points on the sphere, then W1W2 = ?1.
2. The hyperbolic functions sin

Let f be a real function on [a, b]. Suppose that f is Riemann integrable on
[c, b] for every a < c < b.
(a) Show that if f is also Riemann-integrable on
[a, b] then integral b-a(f dx) = limc-a integral b-c(f dx).
(b) Give an example of a function g on [a, b] for which limc-a integral b-c(g dx) is defined, while g is n

Let RI be the set of functions that are Riemann Integrable.
Disprove with a counterexample or prove the following true.
(a) f in RI implies |f| in RI
(b) |f| in RI implies f in RI
(c) f in RI and 0 < c <= |f(x)| forall x implies 1/f in RI
(d) f in RI implies f^2 in RI
(e) f^2 in RI implies f in RI
(f) f^

Complex Differentiation
1) Suppose that an analytic function f defined on the whole of C satisfies Re(f(z))=0 for all z in C. Show that f is constant.
2i) Verify that u=x2-y2-y is harmonic in the whole complex plane.
2ii) Suppose f(x,y)=u(x,y)+iv(x,y). The Cauchy-Riemann equation state that: ux=vy and uy=-vx. For u=x2-