Understanding the Mapping Problem on the Unit Circle Bisected by the X-Axis

[mcafej]

Homework Statement

Take the unit circle in the x-y plane with center at (0, 0), bisected by the x-axis. Take two
maps, the first M_S from the circle minus the south pole S to the x-axis that take a point P on the circle to the intersection of the line from the south pole (0, −1) through P with the x-axis, and the second M_N from the circle minus the north pole N to the x-axis that take a point P on the circle to the intersection of the line from the north pole (0, 1) through P with the x-axis.
(i). Under M_S, what part of the x-axis corresponds to the upper half of the circle? Under M_N , what part of the x-axis corresponds to the upper half of the circle?
(ii). If P = (cos θ,sin θ), show that M_S(P) =cos θ/(1+sin θ),
and M_N (P) =cos θ/(1−sin θ)
.
(iii). If P is any point on the circle other than N or S, show that if M_S(P) = x, then M_N (P) = 1/x

The Attempt at a Solution

I'm really confused on what the actual map is. I tried drawing a circle and seeing what the points would get mapped to, the problem is, what happens to the upper half of the circle under M_N, or the lower half of the circle under M_S? The way I am reading it, the map works by making a circle and then taking a point P on the circle and drawing a line from P to either (0, 1) or (0, -1),depending on if you are using M_S or M_N, and then the point where that line intersects the x-axis is the value that you assign to P. If I am correct about that transformation, that would mean that for the first part of the quesiton, I would get.

i) The upper half of the circle under M_S corresponds the the entire x-axis between the points -1 and 1. Under M_N however, the upper half of the circle does not correspond to anything on the x axis.

Please Help!

 

[micromass]

Homework Statement

Take the unit circle in the x-y plane with center at (0, 0), bisected by the x-axis. Take two
maps, the first M_S from the circle minus the south pole S to the x-axis that take a point P on the circle to the intersection of the line from the south pole (0, −1) through P with the x-axis, and the second M_N from the circle minus the north pole N to the x-axis that take a point P on the circle to the intersection of the line from the north pole (0, 1) through P with the x-axis.
(i). Under M_S, what part of the x-axis corresponds to the upper half of the circle? Under M_N , what part of the x-axis corresponds to the upper half of the circle?
(ii). If P = (cos θ,sin θ), show that M_S(P) =cos θ/(1+sin θ),
and M_N (P) =cos θ/(1−sin θ)
.
(iii). If P is any point on the circle other than N or S, show that if M_S(P) = x, then M_N (P) = 1/x

The Attempt at a Solution

I'm really confused on what the actual map is. I tried drawing a circle and seeing what the points would get mapped to, the problem is, what happens to the upper half of the circle under M_N, or the lower half of the circle under M_S? The way I am reading it, the map works by making a circle and then taking a point P on the circle and drawing a line from P to either (0, 1) or (0, -1),depending on if you are using M_S or M_N, and then the point where that line intersects the x-axis is the value that you assign to P. If I am correct about that transformation, that would mean that for the first part of the quesiton, I would get.

Correct. You might want to search information on "stereographic projections".

i) The upper half of the circle under M_S corresponds the the entire x-axis between the points -1 and 1.

Yes.

Under M_N however, the upper half of the circle does not correspond to anything on the x axis.

No, that is not true. Points on the upper half of the circle certainly do correspond to certain points on the x-axis. For example, P in the picture below certainly does correspond to a certain point on the x-axis, namely Q.

 

[mcafej]

Thank you, that makes a lot more sense. It also helps with part 2 of the problem (where you can take the limit as theta approaches 3π/2 for M_S and you get x>1 or x<1, and you can do the same with M_N, but instead have theta approach ∏/2 to get the same thing).

I am a little confused on how to prove that if M_S(P)=x, then M_N(P)=1/x. Could anybody explain why this is true?

 

[micromass]

Thank you, that makes a lot more sense. It also helps with part 2 of the problem (where you can take the limit as theta approaches 3π/2 for M_S and you get x>1 or x<1, and you can do the same with M_N, but instead have theta approach ∏/2 to get the same thing).

I am a little confused on how to prove that if M_S(P)=x, then M_N(P)=1/x. Could anybody explain why this is true?

I think for that last question, it might be good to give an explicit formulation of the maps M_S and M_N.

So given a point (x,y) on the circle, what is M_S(x,y) in terms of x and y??