#spherical geometry

My second 3D animation!

The orbiting sphere that’s black with a white border is actually a small black sphere closer to the foreground and a white bigger sphere behind it. That was the only solution I could figure out to create that “cel shaded” style. I could have just drawn a 2D ellipse, but since the orbiting planets move through 3D space, the orbit circle would have intersected through the circle.

Learned how to implement spherical geometry on this one! Thanks for the tutorial Daniel Shiffman! Also, this is my first time taking a foray into 3D animations.

The shortest distance between two points on a globe always lies on what they call ‘great circles’. Which is the largest circle you can fit on the sphere, which in other words just means the middle equator part. So like the equator is a great circle, but so are the lines of longitude that all go through the poles. The lines of latitude, though, are not great circles and that’s precisely the point.

The shortest distance between to cities that lie on the same latitude isn’t to go along that latitude, it’s the great circle that goes through the two cities (which doesn’t ever exactly line up with the latitude). So the only straight lines in spherical geometry (since a line is defined as the shortest distance between two points) are great circles!! Which ALSO means that there are no parallel lines in spherical geometry, since every great circle cuts the sphere in exactly half, there’s no way two lines can exist and not intersect.

So there are no parallel lines, straight lines don’t work how you think they should, and triangles all have angles larger than 180 degrees (think about two lines going south from the north pole at 90 degrees from each other and hitting the equator at 90 degrees too. They sketch out a huge triangle with angles 90, 90, and 90!). So it’s weird, but it’s also the actual literal world we live in. All our maps and airplanes and gps’s need to take into account spherical geometry and I think that’s pretty neat.