# Math Help - parametric equations of great circle on sphere

1. ## 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

2. 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 $(\theta, \phi)$ and your sphere has radius R. Then in Cartesian coordinates, the point is $(Rcos(\theta)sin(\phi), Rsin(\theta)sin(\phi), Rcos(\phi))$ and the vector is $Rcos(\theta)sin(\phi)\vec{i}+ Rsin(\theta)sin(\phi)\vec{j}+ Rcos(\phi)\vec{k}$. The cross product of that with the velocity vector is a normal vector to the plane and you already know one point.

3. 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