I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant curvature through out the length as $p(s)=\psi(p_0,{K}_{3 \times 1},s)$, where K is the curvature vector.(You can take it as cosserat rod modeling).
It implies if I take constant $K_0$ as curvature I may get $p(s)$ a different value at s compared when I take $K_1$ for s. Now the question is I am having a curve in the same starting point on same curvelinear length,but its curvature is varying on s as $K_v(s)=(1-s)K_0+sK_1$. What we know is that curve point is some where between $\psi(p_0,{K_0},s)$ and $\psi(p_0,{K_1},s)$. We are equipped with readymade $\psi$.
Question
How can we find the points on new curve whose curvature is a linear interpolation of $K_0$ and $K_1$ from the given $\psi$
NB :: This is not Frenet-Serret Model, it is Cosserat model of curve. The difference is we bend across two cross sectional axices too. Please click here for the pdf where the details of the cosserat curve given at page 4.