Results 1 to 3 of 3

Math Help - parametric equations of great circle on sphere

  1. #1
    Newbie
    Joined
    Jan 2011
    Posts
    2

    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; January 1st 2011 at 07:47 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,971
    Thanks
    1121
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2011
    Posts
    2
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Parametric equations, circle, center, radius
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 9th 2011, 04:42 AM
  2. Great Circle
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 20th 2010, 04:46 AM
  3. parametric equations(movement around a circle)
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 27th 2009, 12:17 PM
  4. Replies: 6
    Last Post: March 28th 2008, 05:19 AM
  5. Replies: 3
    Last Post: October 23rd 2007, 01:44 PM

Search Tags


/mathhelpforum @mathhelpforum