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.
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 half-circle 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.