Interpolate the orbital coordinates of an object using coordinates and velocities vector

Question

I have a table of the orbital coordinates and velocities of an object with time steps of 1 minute.

Now I would like to interpolate this to a finer time increments with time steps of the order of 1 second. How to make use of the velocities in coordinate interpolation and what might be the most accurate way to do this?

Answer 1

You can interpolate values between $[a, b]$ using Hermite interpolation. You first map the interval $[a, b]$ to $[-1,1]$, and the values are computed as: $$f(x) \approx N_1(x) f(a) + N_2(x) f(b) + \frac{b - a}{2}[N_3(x) f'(a) + N_4(x) f'(b)]\quad \forall x\in [-1, 1]$$

with \begin{align} N_1 (x) &= \frac{1}{4} (x - 1)^2 (2 + x)\\ N_2 (x) &= \frac{1}{4} (x + 1)^2 (2 - x)\\ N_3 (x) &= \frac{1}{4} (x - 1)^2 (x + 1)\\ N_4 (x) &= \frac{1}{4} (x + 1)^2 (x - 1)\, . \end{align}

Then, you just need to loop over each pair of points.

Answer 2

That depends on how well you know the coordinates and velocities. If you have exact values, you can get a reasonable answer using Hermite interpolation. This will give you a degree-3 polynomial in each window that matches the coordinates and velocities at all the endpoints.

Alternatively, if you do not know the coordinates and velocities exactly, but you do know something about the probability distribution of the measurement errors, you'll want to use Kalman smoothing or a similar algorithm. Kalman-type methods are especially nice because you can also incorporate the physics into how you interpolate the data.