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Math Help - 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 \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 \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 \phi with the xy-plane and that the line between the two intersections with the xy-plane makes angle \psi with the x-axis. Let \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 \phi= 0) The points on that circle are given by x= cos(\theta), y= sin(\theta), and z= 0. Now we can just "rotate" that to angle \phi. z= r sin(\phi)[/tex] where r is ysin(\psi)= sin(\psi)sin(\theta): z= sin(\theta)sin(\psi)sin(\phi) so that the parametric equations for the circle at fixed angle \phi with the xy-plane and fixed angle \psi with the x-axis, with parameter \theta are
    x= cos(\theta), y= sin(\theta), z= sin(\theta)sin(\psi)sin(\phi).
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