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Caclulus Mapping problem

  1. Dec 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Take the unit circle in the x-y plane with center at (0, 0), bisected by the x-axis. Take two
    maps, the first MS 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 MN 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 MS, what part of the x-axis corresponds to the upper half of the circle? Under MN , what part of the x-axis corresponds to the upper half of the circle?
    (ii). If P = (cos θ,sin θ), show that MS(P) =cos θ/(1+sin θ),
    and MN (P) =cos θ/(1−sin θ)
    .
    (iii). If P is any point on the circle other than N or S, show that if MS(P) = x, then MN (P) = 1/x


    3. 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 MN, or the lower half of the circle under MS? 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 MS or MN, 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 MS corresponds the the entire x axis between the points -1 and 1. Under MN however, the upper half of the circle does not correspond to anything on the x axis.

    Please Help!!!!
     
  2. jcsd
  3. Dec 3, 2012 #2

    micromass

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

    Yes.

    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.

    image002.gif
     
  4. Dec 3, 2012 #3
    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 MS and you get x>1 or x<1, and you can do the same with MN, but instead have theta approach ∏/2 to get the same thing).

    I am a little confused on how to prove that if MS(P)=x, then MN(P)=1/x. Could anybody explain why this is true?
     
  5. Dec 3, 2012 #4

    micromass

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

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