Say I have two parametric tubes of the form $C_i(u,v) = (x(u, v), y(u, v), z(u,v))$. The most common case would be sweeping surfaces along a path, but the sweep can have a scaling factor.
These functions are commonly used in computer graphics and CAD software.
I am curious as to how one could merge them while interpolating their endpoints. For example, say that the tubes are parametrized such that the curves $C_i(u,v_0)$, $C_i(u,v_1)$ form the boundaries of the tubes. That is, they are closed curves and they form the ridge at the end of the shape.
Each curve has a different swept profile, i.e. both $C_1$ and $C_2$ are closed planar curves embedded in $\mathbb{R}^3$. An order one stitching is trivial, just arc length parametrize both of $C_1, C_2$ and linearly interpolate, i.e.:
$$ C_1(u, v_1) (1 - t) + C_2(u, v_0) t$$
Is there literature anywhere on how to achieve higher order stitching of these shapes?