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Math Help - Help finding 90 degree angle in sphere

  1. #1
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    Help finding 90 degree angle in sphere

    Hello,

    Struggling artist here, please excuse my very basic understanding / lack of technical terms. I've included a quick diagram to make it easier to explain my question.

    Is it possible, assuming distance "a" is known, to determine what distance "b" will be? I'm trying to better understand how far apart 90 degree slices are when rotating a spherical object, but obviously the distance changes depending on where the two points are on the rotating axis relative to the viewpoint. Does that make sense?

    (also, in the diagram,, by "b" i mean distance on the centre line).

    Thank you!
    Attached Thumbnails Attached Thumbnails Help finding 90 degree angle in sphere-photo-2-.jpg  
    Last edited by sneakypete; December 7th 2011 at 10:13 AM. Reason: typo
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  2. #2
    Senior Member
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    Re: Help finding 90 degree angle in sphere

    Are you saying that 6 is "a" and "b" is the distance moved, i.e. 9?
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  3. #3
    Junior Member
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    Re: Help finding 90 degree angle in sphere

    Interesting geometry problem. I'm assuming that you mean that a and b are lengths on a two-dimensional photograph of the sphere (what we in math jargon would call the projection of the sphere onto a plane). In that case...

    Let's define \theta as the rotation angle of the sphere. Let \theta=0^{\circ} correspond to the situation where a equals the sphere's radius and b is zero. If the rotation angle \theta is increased, then a decreases (and b increases) according to:

    a=R\sin(90^{\circ}-\theta)
    b=R\sin(\theta)
    in which R is the radius of the sphere.

    So if you know distance a then you calculate the rotation angle \theta which you can use in turn to calculate b. A quick rewrite of the formulas given for this:
    \theta=90^{\circ}-\arcsin\left(\frac{a}{R}\right)
    b=R\sin(\theta)

    Let us know if this gives you the results that you were expecting.
    Last edited by corsica; December 7th 2011 at 11:28 AM.
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  4. #4
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    Re: Help finding 90 degree angle in sphere

    Perfect, thank you corsica.
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