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Thread: parametric equations of great circle on sphere

  1. #1
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    parametric equations of great circle on sphere

    Hello all,
    I'm new to the forums and I hope this is the appropriate subforum for this question.
    I'm programming a sphere that rolls around in straight paths. for the particular library i'm using, i essentially indicate phi/theta and it essentially returns 2D graphics of a sphere rotated to the given angles.
    My goal is to rotate the sphere starting at a given starting angle and every frame i would supply the new spherical coordinates. thus i would need to have phi and theta as a function of time for a regular rolling ball.

    so the question essentially:
    what are the equations for phi and theta of a great circle in parametric form, specifically, in terms of time given a starting phi/theta and a velocity?

    I've been juggling a few ideas in my head but this is really not my forte. any help would be appreciated

    edit underlined
    Last edited by zyxstand; Jan 1st 2011 at 07:47 PM.
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  2. #2
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    If you are "given a velocity" then you are given a direction- a vector tangent to the sphere in the direction of motion. Every great circle is the intersection of the sphere with a plane through the center of the sphere. You can think of that plane as generated by two lines- the line from the given point to the center of the sphere and the line in the same direction as the velocity vector.

    Let's say you are given $\displaystyle (\theta, \phi)$ and your sphere has radius R. Then in Cartesian coordinates, the point is $\displaystyle (Rcos(\theta)sin(\phi), Rsin(\theta)sin(\phi), Rcos(\phi))$ and the vector is $\displaystyle Rcos(\theta)sin(\phi)\vec{i}+ Rsin(\theta)sin(\phi)\vec{j}+ Rcos(\phi)\vec{k}$. The cross product of that with the velocity vector is a normal vector to the plane and you already know one point.
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  3. #3
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    thanks! that makes sense. I do still have two problems:
    1. I'm working with spherical coordinates, not Cartesian. It seems more of a hassle converting to Cartesian, solving, and going back to spherical coordinates.
    2. I don't know how I could convert the new equation to a set of time-dependent parametric equations
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