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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?

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If I'm understanding you correctly, you define a quaternion $q(x,y)$ between two points $x,y$ lying on a space curve by taking the change in orientation between the Frenet frames at those points, $\tilde{q}(x,y)\in SO(3)$ and lifting it to the double cover $\operatorname{Spin}(3)$, which is isomorphic to the set of unit quaternions.

[Note that you must assume that the first and second derivatives of the unit tangent vector of the curve at $x,y$ are linearly independent, otherwise the Frenet frame is not well-defined].

Getting a well-defined quaternion requires some more choices on your part. One way to proceed is define $q(x,y)$ as the endpoint ($t=1$) of the continuous lift of the curve $C_{xy}=\{\tilde{q}(x,x(t))|t\in[0,1]\}\subset SO(3)$, where I assume that $x(0)=x$, $t=1$ is the smallest value of $t$ such that $x(t)=y$ and that the nondegeneracy condition holds for all $x(t)$ with $0\leq t\leq 1$.

Your first question seems to be asking for the value of $\lim_{y\rightarrow x}q(x,y)$ (wherever it is defined). However, note that $\tilde{q}(x,x)$ is the identity! For my definition above, $q(x,x)$ is also the identity, since the curve $C_{xy}$ degenerates to a point.

If you want something nontrivial, you can look at the derivative with respect to the arclength. Then there is a nontrivial element of $so(3)$, the Lie algebra of $SO(3)$ (and hence of Spin(3)) at nondegenerate points of a space curve. It can be expressed in terms of the curvature and torsion, per the Frenet-Serret formulas.

Similarly, the set of quaternions you ask about in the surface case again seems to consist of only the identity, no matter the direction of approach. Looking at derivatives, you get the elements of $so(3)$ given by the formulas for the Darboux frame.

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