The azimuthal equidistant map projection can easily be applied to any 2D manifold. Are any other projections so generally applicable? Probably many azimuthal ones.
From the center point, the map extends in a given straight-line direction, traveling along a geodesic, until it encounters points for which that geodesic ray isn't the shortest path to those points. (Instead, perhaps go around the other way.) Can really weird manifolds have concave maps?
There are probably also regions unreachable by any geodesic from some starting points.
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