I would like to define a rotation minimizing frame (also known as a Bishop Frame) for a curve. This should be equivalent to parallel transporting a unit normal vector along the curve and then building a frame with the unit tangent vector. I've read of approximate methods for doing this (e.g., "Computation of rotation minimizing frames" [Wang et al. 2008]).
I'm wondering if it's known how to express the Bishop Frame in closed form, if we specifically consider a cubic Bézier curve in 3D:
$$ \mathbf{x}(s) = (1-s)^3 \mathbf{p}_0 + 3(1-s)^2s \mathbf{p}_1 + 3(1-s)s^2 \mathbf{p}_2 + s^3\mathbf{p}_3, $$ where the control points $\mathbf{p}_i \in \mathbb{R}^3$ are constants?
(Or whether it's known that a closed form somehow cannot exist?)
If not, I'm hoping for some pointers on how to derive this (I am happy to use symbolic math software, too).
So far, I've had some luck with Maple and quadratic Bezier curves by hardcoding a mapping of the curve to the $xy$ plane and fixing $\mathbf{p}_0$ at the origin and $\mathbf{p}_1$ at $(1,0,0)$, taking advantage of similarity invariance of parallel transport. Then I tell Maple to solve the ODE:
$$\mathbf{f}(0) = \mathbf{f}_0, \mathbf{f}'(s) = (\mathbf{t}(s) \times \mathbf{t}'(s)) \times \mathbf{f}(s),$$ where $\mathbf{t}(s) = \mathbf{x}'(s)/\|\mathbf{x}'(s)\|$ is the unit tangent of my cubic Bézier curve. Maple's crunching away at the cubic version, but so far nothing.
Is this the best/easiest ODE to be using for parallel transport? Is there some more convenient expression that would reveal a closed form expression?