Given a portion of a sphere strictly less than a hemisphere, project it onto a plane minimizing the maximum distortion anywhere, for some appropriate measure of distortion. Probably an underspecified problem.
Reverse: given a flat image, project it onto (at most) a hemisphere, minimizing distortion.
Do both in sequence (reverse, then forward). Original motivation: slightly distorted QR codes.
Ellipsoid, hyperboloid, hyperbolic paraboloid, tractricoid (pseudosphere). Hyperbolic plane (do we need to specify an embedding (immersion?) in 3-space to define distortion?).
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