Reminds me of a complex logarithm. But that function has infinite extent and countably infinite turns. And yet I see your conformal manifold wrap around infinity and the geometric structure just resonates. How are you mathematically representing these?
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They use reverse stereographic projection to the 3-sphere, then a 4d rotation, then projection back. First to sweep out a surface, then again to create the animated transformation.
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..also I should add - when sweeping out the surface, the line is simultaneously rotating in Euclidean space, which gives it an extra twist. Otherwise the surface would just slide along itself in the animated transformation
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