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Math Help - Cartesian to Spherical

  1. #1
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    Cartesian to Spherical

    Hi,

    I need to convert these coordinates from cartesian to spherical.

    (3,4,12)

    and these coordinates from spherical to cartesian

    (5, \frac{\pi}{4}, \frac{\pi}{6})

    I've done a bunch of searches, but none of the websites I found were very clear. Could someone explain this to me?
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  2. #2
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    You should know that

    x = r\sin{\theta}\cos{\phi}

    y = r\sin{\theta}\sin{\phi}

    z = r\cos{\theta}.


    Going the other way:

    r = \sqrt{x^2 + y^2 + z^2}

    \theta = \arccos{\frac{z}{x^2 + y^2 + z^2}}

    Finding \phi is more difficult as it depends on the values of x and y.

    Namely:

    \phi = \arctan{\frac{y}{x}} for x > 0

    \phi = \pi + \arctan{\frac{y}{x}} for y \geq 0, x < 0

    \phi = -\pi + \arctan{\frac{y}{x}} for y < 0, x < 0

    \phi = \frac{\pi}{2} for y > 0, x = 0

    \phi = -\frac{\pi}{2} for y < 0, x = 0

    \phi = undefined for y = 0, x = 0.


    These values come from elementary trigonometry.
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