How to make smooth spline interpolation setting derivatives only at specific points, in Python?

Question

TL;DR

Apparently scipy doesn't provide a method similar to MATLAB's spapi that allows for setting the derivatives only at specific points, ensuring smooth transitions all along the interpolated spline. I'd like to find a way to do that in Python.

Overall

My goal is to make spline interpolation using Python 3, but setting the desired derivatives only at specific points. For example, say the array X defines position of an object, point to point, I'd want a spline to represent a feasible trajectory, but also ensuring that the speed (first derivative) is zero at both starting and ending points, other points having no constraints on the derivatives. In this example, I could also desire to have the acceleration (second derivative) equal to zero at the same points.

What I want is, in other words, an implementation of spline interpolation similar to the MATLAB's spapi function, as referenced here:

spline = spapi(knots,x,y) returns the spline f (if any) of order

k = length(knots) - length(x)

with knot sequence knots for which

(*) f(x(j)) = y(:,j), all j.

If some of the entries of x are the same, then this is taken in the osculatory sense, i.e., in the sense that Dm(j)f(x(j)) = y(:, j), with m(j) : = #{ i < j : x(i) = x(j) }, and Dmf the mth derivative of f. Thus r-fold repetition of a site z in x corresponds to the prescribing of value and the first r – 1 derivatives of f at z.

What I've tried

I know of the method from_derivatives of the BPoly class of scipy.interpolate, but that poses a critical problem, being that when derivatives are not specified at any point, the algorithm does not guarantee smooth transition, as noted here. I also tried what was proposed here, but as expected, the same problem arises.

So, next I present simple reproductions of what I'm trying to achieve, successful or not.

spapi without derivatives

Here, example using spapi method, in MATLAB, without setting any derivatives. Not as what I want since the derivatives are high at start and end points:

xi = linspace(0, 10, 6);
xnew = linspace(0, 10, 100);

yi = [0 2 1 4 2 0];

knots = optknt(xi, order);
ref_spline = spapi(knots, xi, yi);

spline = fnval(xnew, ref_spline);
der_spline = gradient(spline);

And the corresponding plot of reference points along with the interpolated spline:

And here the 1st order derivative of that spline:

spapi with derivatives

Here, example using spapi method, in MATLAB, setting derivatives as 0 at both start and end points. Exactly the expected result, with smooth transitions all along the spline, and derivatives equal to 0 at start and end points:

xi = linspace(0, 10, 6);
xnew = linspace(0, 10, 100);
xder = [0 xi 10];

yi = [0 2 1 4 2 0];
ynew = [0 yi 0];

knots = optknt(xder, order);
ref_spline = spapi(knots, xder, ynew);

spline = fnval(xnew, ref_spline);
der_spline = gradient(spline);

And the plot of reference points along with the spline:

And here the 1st order derivative of that spline:

BPoly.from_derivatives with derivatives

Here, example using BPoly.from_derivatives setting derivatives as 0 at both start and end points. Unsuccessful, even though the derivatives are 0 at start and end, smooth transition is not guaranteed along the spline:

ref_points = [0, 2, 1, 4, 2, 0]
time_vector = np.linspace(0, 10, 100)
time_points = np.linspace(0, 10, 6)

ref_complete = [[ref_points[j] if (i == 0) else 0 for i in range(2)] if (
    (j == 0) or (j == len(ref_points) - 1)) else [ref_points[j]] for j in range(len(ref_points))]

ref_spline = BPoly.from_derivatives(time_points, ref_complete)
spline = ref_spline(time_vector)

der_spline = ref_spline.derivative(1)
der_y = der_spline(time_vector)

To clarify, the line that defines ref_complete simply replaces the first and last elements of ref_points by an array containing the original value at index 0, and a 0 at index 1:

>>> ref_points
[0, 2, 1, 4, 2, 0]
>>> ref_complete
[[0, 0], [2], [1], [4], [2], [0, 0]]

And the plot of reference points along with the spline:

And here the 1st order derivative of that spline:

Answer 1

If you just want to do cubic spline interpolation with zero derivatives at both ends this is enough (docs here)

from scipy.interpolate import CubicSpline

CubicSpline(time_points, ref_points, bc_type="clamped")