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I am aware the two frames are different, aside from sharing the same tangent unit vector in their basis. But I was wondering, why or when would one choose to use/work with one other frame and not the other? What information does one of frame have that the other does not?

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The Frenet frame is used to study curves, while the Darboux frame is used to study curves in surfaces. In general, frames should be adapted to whatever geometric object you're trying to study. This is the reason why both frames always have the velocity vector of the curve in it.

A priori, if you rotated the frame vectors in the normal plane to the velocity vector together there, you would a legitimate frame to study the curve. However, the Frenet frame is special because it is constructed from the other derivatives of the curve itself.

Now, if your curve lies in a surface, the frame should be adapted not only to the curve, but to the surface as well, and that is the reason why not only the velocity vector to the curve is in your Darboux frame, but also the normal vector to the surface as well. Since we look for orthonormal frames, these requirements determine the last vector in the frame.

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