Fun with Spheres pt 2

My previous explorations really taught me that I had absolutely NO CLUE how to spherical geometry, so I’ve spent these past weeks teaching myself how to get around in spherical coordinates. I’ve  come up with a potpourri of sketches exploring spherical geometry. These are mostly for my own benefit, but I thought I’d share anyways.

Oct29-1231          This sketch does the simplest things, like drawing great circles and small circles. It was a very good primer to start thinking in a spherical mindset. The strategy at this point was to always imagine important vectors being aligned along the XYZ axes, and which reduced the problem down to a much simpler one, and then implementing the solution through rotation transformations.

 
 
 
 

Nov2-0398          I wondered what kind of path a point on the sphere would take if it was constantly rotating around some moving axis. To this end, Rodrigues’ rotation formula proves extremely helpful.
 
 
 
 
 
 
 
 
 
Nov2b-0465          Going back to my roots – gravitation! The red ball is gravitationally attracted to the green balls using a spherical distance metric. This was actually tricky because velocities aren’t as straightforward as they are in Euclidean space. I ended up creating the notion of a “Local Plane”, which is basically an azimuthal projection of the sphere centered at the red ball. This lets you do vector math as if you were in a simple 2D space, so it’s very easy to add forces and integrate for velocity.
 
 
 

I also re-wrote FlexiLine on the spherical plane. See the sketch here.

Nov9-0783


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One Comment on “Fun with Spheres pt 2”

  1. Brendan ☼ says:

    Sweet stuff mate.

    Regarding planets, spheres and gravity, I thought this might be of interest: http://www.straightdope.com/columns/read/3011/what-would-it-be-like-walking-around-on-a-cube-shaped-planet

    The article reminded me of two of Escher’s woodcuts: Double Planetoid and Tetrahedral Planetoid:
    http://cargocollective.com/escher#613/Dubbele-Double-Planetoid
    http://www.wikipaintings.org/en/m-c-escher/tetrahedral-planetoid


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