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.
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.
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.
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.