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Math Help - Function to map an annulus to a circle?

  1. #1
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    Function to map an annulus to a circle?

    I need to describe a function which maps an annulus to its 'inner' circle.
    Like, for mapping the torus X to the annulus Y, I said that for every point x\in X with coordinates (i,j,k) on the torus, f(X):x\rightarrow y| y_i=x_i, y_j=x_j, y_k=0.. I'm looking for something similar to then map the ys onto a circle, by "contracting" it, as it were.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Are you wanting a nice, smooth map? A bijection? Or just some function?

    I can't think of a neat one - I'm sure a neat one will exist! However, assume your annulus is centered at the origin. Then you know that you can take one closed interval [a, b] to another [c, d] by the function x \mapsto \frac{x-a}{b-a}(d-c)+c. So, generalise this. Take the point (x, y) in your annulus and simply map it so far along the line (that is, on the positive part of the x-axis you will just be moving the interval [a, b] to [0, a], and just rotate this all the way around).

    I will leave you to fill in the details.
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