Purchase Solution

Stereographic projection on complex plane

Not what you're looking for?

Ask Custom Question

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_3)^2=1. Use this information to show that if V contains the point N then its seteographic projection on the complex plane is a straight line. Otherwise, V projects onto a circle in complex plane.

N = (0,0,1) the north pole on S
Please explain to me every step, I don't just want the answer, but I also want to understand it and be able to work similar problems.

Purchase this Solution

Solution Summary

A stereographic projection on complex plane is investigated.

Solution Preview

Proof:

V is a circle. V determines a unique plane P in R^3 and V lies in this plane. So P^S=V.
We consider two cases.
Case 1: V contains N=(0,0,1). Since V=P^S, then N is also on the plane P. The equation of P is x_1*b_1+x_2*b_2+x_3*b_3=L, where (b_1)^2+(b_2)^2+(b_3)^2=1. We plug in (0,0,1) and get b_3=L. We also know that the center of the sphere S is (0,0,0) and this center is also the center of V. So we have L=0. ...

Purchase this Solution


Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.