How to calculate the orientation of an arc on a sphere?

If you want to do it all in After Effects, I would approach the problem this way:

1. Import your 3D Earth model (obj) and add it to an Advanced 3D composition
2. Create a 3D null and Shift + Parent the Null to the Earth model so that it is in the center of the earth
3. Create a 3D shape layer with No Content yet
4. Shift + parent the Shape layer to the Center 3D Null
5. Use the Pen Tool or the Ellipse tool to create an arc on the edge of the sphere, extrude it, and bevil it a bit
6. Rotate the Center 3D null to position the arc over the points on the earth
7. Change the first Arc Parent to the Earth layer, then repeat the process for another arc until you have your shapes all lined up with the Earth model

You will not be able to end up with round-looking lines if you stick to After Effects. If you stick with the Classic 3D rendering engine, you will end up with flat lines.

You could accomplish the same thing and get a better-looking result if you combined some 3rd Party effects like Element 3D or VC Orb from Video Copilot, 3D stroke from Red Giant/Maxon, or even C4D lite that comes with After Effects. If you want to step completely into creating the effect in a 3D app, try Blender (http://blender.org). It's very compatible with After Effects, and a great set of tutorials are available on the site.

Could be as trivial as adding a lookAt() expression towards the center of the sphere to your 3D layers... Of course there's other, more complex ways, too, if needed.

Mylenium