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Thread: 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, $\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{\pi}{6}$)

    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

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

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

    $\displaystyle z = r\cos{\theta}$.


    Going the other way:

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

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

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

    Namely:

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

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

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

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

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

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


    These values come from elementary trigonometry.
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