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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.

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  • $\begingroup$ 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 Edgar Sep 15 '17 at 11:58
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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.

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