A transformation of a surface is "conformal" if at each point the transformation preserves angles. (In a bit more detail, if we pick an arbitrary point and pick two curves meeting perpendicularly at that point, then the image curves meet perpendicularly, and with the "same handedness" as the original curves.) The most obvious conformal transformations also preserve lengths, such as translations and rotations of the Euclidean plane. (Reflections preserve perpendicularity, but "reverse" handedness, so are not conformal in the sense meant here.)
A conformal transformation of a sphere is a "Möbius transformation." It might appear that all Möbius transformations are rotations, but in fact there are many others. The spiral curves here are "flow lines" of a "one parameter group" of Möbius transformations.
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