Optical mirror for cylindrical to planar reflection

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I was thinking of converting a disused chimney into a static periscope, using long focal length lenses at top end of the chimney. A static design would be easier, so I was wondering how one would design the mirror shape such that it converted a cylindrical view onto the lens, which could then be processed at the other end of the chimney to generate a cylindrical view once more.

Is it even possible?