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Thread: Great Circle

  1. #1
    MHF Contributor Swlabr's Avatar
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    Great Circle

    Hi,

    Can anyone please tell me the equation of a great circle on the sphere in radius 1 in $\displaystyle \mathbb{R}^3$ in terms of sines and cosines?

    Thanks.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Swlabr View Post
    Hi,

    Can anyone please tell me the equation of a great circle on the sphere in radius 1 in $\displaystyle \mathbb{R}^3$ in terms of sines and cosines?

    Thanks.
    Any "equation" for a geometric figure depends on how you set up the coordinates system so I am going to assume your sphere is at the origin of the coordinate system.

    Any great circle? Okay, assume the great circle makes angle $\displaystyle \phi$ with the xy-plane and that the line between the two intersections with the xy-plane makes angle $\displaystyle \psi$ with the x-axis. Let $\displaystyle \theta$ be the angle a line from the center of the sphere (the origin) to intersection of the sphere with the xy-plane (the circle with $\displaystyle \phi= 0$) The points on that circle are given by $\displaystyle x= cos(\theta)$, $\displaystyle y= sin(\theta)$, and z= 0. Now we can just "rotate" that to angle $\displaystyle \phi$. z= r sin(\phi)[/tex] where r is $\displaystyle ysin(\psi)= sin(\psi)sin(\theta)$: $\displaystyle z= sin(\theta)sin(\psi)sin(\phi)$ so that the parametric equations for the circle at fixed angle $\displaystyle \phi$ with the xy-plane and fixed angle $\displaystyle \psi$ with the x-axis, with parameter $\displaystyle \theta$ are
    $\displaystyle x= cos(\theta)$, $\displaystyle y= sin(\theta)$, $\displaystyle z= sin(\theta)sin(\psi)sin(\phi)$.
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