How to find rotation parameter in sphere?

[rahman_f_t]

Suppose there is one circle ( signal shape ) on the sphere. then the sphere is rotate. By comparing these two circle, is it possible to find 3 rotation parameters or angles in Euler coordinate (or 2 rotation angle, sphere coordinate)?
Any idea?
Thanks

 

[HallsofIvy]

One obvious difficulty is that, if the axis of rotation is perpendicular to the circle, the circle will rotate into itself. Clearly, just knowing the circle will not tell us the angle the sphere rotated.

 

[rahman_f_t]

One obvious difficulty is that, if the axis of rotation is perpendicular to the circle, the circle will rotate into itself. Clearly, just knowing the circle will not tell us the angle the sphere rotated.

Suppose it is not really circle, there are curve on it ( like as wave signal on the circle). how about this time?