Interpolate between N points a curve that is 6 times differentiable and that is subject to boundary conditions for each derivative

Question

I have a set of points $\{(x_1,y_1),...,(x_N,y_N)\}$ and I want to interpolate a curve that is at least $6$ times differentiable. Moreover, at the border, so for the points $(x_1,y_1)$ and (x_N,y_N) the derivative up to the $6$th have to be equal a specific value that I know in advance. For the remaining points the derivatives are free. A solution would be to define $N$ polynomials $$y=g_i(x)={}_ia_0+{}_ia_1x+{}_ia_2x^2+...+{}_ia_{2N-1}x^{2N-1} $$ for $x\in(x_i,x_{i+1})$ and corresponding conditions to ensure the continuity and values on the borders. Is there a different method? Ideally the total length of the curve is minimized. In a next step, the set will be constituted of points from $\mathbb{R}^3$ so $\{(x_1,y_1,z_1),...,(x_N,y_N,z_N)\}$. So the method should be easily extensible.