Can we introduce independent coordinates on a sphere such that any great circle could be represented as a linear equation (like line on the plane)? If yes, what is a generalization for higher dimensions? Thank you in advance.

$\begingroup$ When you say "multidimensional" are you then talking about great hypercircles (or whatever they would be called)? The intersection of a hyperplane with the sphere. $\endgroup$– Gerald EdgarSep 15, 2017 at 11:58
1 Answer
This can be done locally by projection from the center of the sphere. Great circles go to lines, so if you pull back the cartesian coordinates from the plane to a hemisphere, every great halfcircle is described by a linear equation.
You cannot extend this beyond the hemisphere, because two great circles meet in two points, but a system of linear equations cannot have two solutions.