In 3D differential geometry there are canonical local coordinate systems associated with each point of sufficiently smooth curves and surfaces:

in the case of curves there are the Frenet frames $$\frac{\dot{x}(t)}{\|\dot{x}(t)\|},\frac{\ddot{x}(t)}{\|\ddot{x}(t)\|},\frac{\dot{x}(t) \times\ddot{x}(t)}{\|\dot{x}(t) \times\ddot{x}(t)\|}$$

in the case of surfaces we have, except at umbilical points, the mutually orthogonal directions of extremal normal curvature and the surface normal

and in both cases the orthogonal frames of two locations correspond to a rigid motion, which in turn can be expressed by a rotation about an angle $\alpha$ around a vector or, alternatively by a quaternion and the addition of a vector.

Questions:Is there a name for the quaternion to which the "quaternionic part" of the rigid motion of a curve's frenet frame converges as its origin approaches $x(t)$? In the case of surfaces there is an entire set of quaternions associated with each point; is there also a name for that set?

In the case of surfaces: what can be said about the rotation-part of the quaternions when approaching a surface point from different directions in the $uv$-coordinate plane; is $\alpha(u,v)$ continuous and how is its smoothness related to the surface's smoothness?