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Thread: Stereographic Projection problem

  1. #1
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    Stereographic Projection problem

    Show that stereographic projection takes parallels of latitude (phi) on the sphere (unit sphere S2), given as z = sin (phi), into the circles in complex numbers with centre O and radius
    1 + tan(phi/2)
    1 - tan(phi/2)

    Please help... I despise projections with a passion and I am not entirely sure where to even begin with this problem
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  2. #2
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    Re: Stereographic Projection problem

    well... I'm reading that the projection is the surface of a sphere to a plane given by

    $sp(x,y,z) = \left \{ \dfrac{x}{1-z},~\dfrac{y}{1-z}\right \}$

    let $\phi$ be latitude, and $\theta$ be longitude. Let the radius of the earth be 1.

    $x = \cos(\phi)\cos(\theta)$

    $y = \cos(\phi)\sin(\theta)$

    $z = \sin(\phi)$

    $sp(\phi, \theta) = \left \{ \dfrac{\cos(\phi)\cos(\theta)}{1-\sin(\phi)},~\dfrac{\cos(\phi)\sin(\theta)}{1-\sin(\phi)}\right \}$

    $sp(\phi, \theta) =\dfrac{\cos(\phi)}{1-\sin(\phi)}\left \{ \cos(\theta),~\sin(\theta)\right \}$

    for a fixed $\phi$ this is clearly a circle as $-\pi < \theta \leq \pi$

    it's radius is just the factor to the left

    if you muck about with this a bit you can obtain the expression they give you for the radius though I don't know why this expression isn't good enough.
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  3. #3
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    Re: Stereographic Projection problem

    Thank you so much! Do you have website you can direct me to that has these formulas you used on it?
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  4. #4
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    Re: Stereographic Projection problem

    Quote Originally Posted by randarae View Post
    Thank you so much! Do you have website you can direct me to that has these formulas you used on it?
    the first was just the wiki on stereographic projection

    the conversion to spherical coordinates is just common math knowledge. There's probably a wiki on "spherical coordinates"
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