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Math Help - Proof: Arc length across a sphere in spherical coordinates

  1. #1
    Junior Member beebe's Avatar
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    Proof: Arc length across a sphere in spherical coordinates

    Use spherical polar coordinates (r, /theta, /phi) to show that the length of a path joining two points on a sphere of radius R is  L=R\int_{\theta_1}^{\theta_2}\sqrt{1+sin^2\theta\p  hi'(\theta)^2}d\theta

    I was initially told that phi should be rotation around the z-axis measured from the x-axis, and theta was the angle up out of the x-y plane. Working it this way I set up a triangle with hypotenuse dL and legs Rd\theta and |\overrightarrow{r}_{proj}|d\phi. The following picture shows how I have the coordinates set up:

    Proof: Arc length across a sphere in spherical coordinates-path-problem.png
    This gives me:

    |\overrightarrow{r}_{proj}|=Rcos\theta

    dL^2=R^2d\theta^2+|\overrightarrow{r}_{proj}|^2d\p  hi^2=R^2d\theta^2+R^2cos^2\thetad\phi^2

    L=\int_{1}^{2}dL=R\int_{\theta_1}^{\theta_2}\sqrt{  d\theta^2+cos^2\theta\phi'(\theta)^2d\theta^2} = R\int_{\theta_1}^{\theta_2}\sqrt{1+cos^2\theta\phi  '(\theta)^2}d\theta

    Which is what I was supposed to get except I have cosines instead of sines. This problem is easily solved by having \theta be measured down from the z-axis instead of up from the x-y plane, but someone had told me to set up the coordinates as I specified above. I was hoping someone could either verify that what I've done is correct or point out where I messed up. Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Proof: Arc length across a sphere in spherical coordinates

    The typical way to define theta is to measure it down from the +z axis. There may be some special cases where you wouldn't do that, but down from the +z is standard.

    -Dan
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  3. #3
    MHF Contributor

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    Re: Proof: Arc length across a sphere in spherical coordinates

    That's the "British" system which is reverses the angles: in the American system, \theta is the angle measured around the z- axis, \phi is the angle measured from the z-axis.
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  4. #4
    Junior Member beebe's Avatar
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    Re: Proof: Arc length across a sphere in spherical coordinates

    I was looking for verification that the trig function that I got in my result is correct, that is, that I should have cos if measuring up and sin if measuring down. Although I think I got the answer, my book had a similar problem in which the angle was measured down from the z axis, which would give me sin, which is consistent with what they were asking me to prove. Thank you for the responses.
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