Interpolation methods for two points and three derivates?

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Discussion Overview

The discussion revolves around interpolation methods for estimating the minimum distance between two particles, given their positions, velocities, accelerations, and jerks at two time points. Participants explore the applicability of various interpolation techniques, particularly in the context of including higher-order derivatives.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the effectiveness of Piecewise Cubic Hermite Interpolation, suggesting it may not be the most accurate method due to its reliance on only positions and velocities.
  • Another participant notes that knowing the positions and derivatives at two points provides enough information to fit a high-degree polynomial, specifically a 23rd degree polynomial.
  • A different viewpoint expresses skepticism about determining the minimum distance, arguing that the behavior of the particles between the two time points is uncertain.
  • Concerns are raised regarding the necessity of having n-1 points to fit an nth order polynomial and the potential artifacts from high-order polynomial interpolation.
  • One participant suggests Hermite Interpolation as a method that incorporates both data points and their derivatives, seeking further resources for a deeper understanding of this method.
  • A later reply mentions that the context involves central forces and Keplerian orbits, implying that the higher derivatives may not exhibit extreme behavior.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of various interpolation methods and the implications of including higher-order derivatives. There is no consensus on the best approach or the feasibility of determining the minimum distance accurately.

Contextual Notes

Participants highlight limitations related to high-degree polynomial fitting, potential artifacts, and the uncertainty of particle behavior between the specified time points.

theturbanator
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Hi everyone,

I have a question about interpolation methods. I am given two particles and I know their positions, velocities, accelerations, and jerks (time derivative of acceleration) at some initial and final time-values (t0 and t1, respectively). I want to find the minimum distance between them in the range [t0,t1]; thus I need to do an interpolation. I have seen a method of Piecewise Cubic Hermite Interpolation used; however, it only uses the positions and velocities of the particles and is, therefore, not the most accurate predictor of the minimum distance. Is there an interpolation method that can be applied to two points and the three derivatives at those points that is better than the above method?

Thank you for all of your help.
 
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You are essentially saying that you know 8 3-vector quantities at two different times. That means that you know 48 values for the two particles or 24 for each one. You can fit a 23rd degree polynomial to those.
 
Well, you can't know the minimum possible, because anything could have happened in-between the start and final points.

If you just want to estimate the path of each particle, why not just assume the higher derivatives are all constant?
 
There are two concerns I have about that, though:

first, I thought that in order to fit a nth order polynomial, you need n-1 points?

second, polynomial interpolation of such a high order gives horrible artifacts; is there some way to keep order of the polynomial (in terms of the time variable) low, but to include the higher-order derivatives? One thing that doesn't make sense to me is that if I have more constraints on the initial and final state of the particles (because I am including the initial and final accelerations and jerks), shouldn't the interpolated path (remember I am interpolating the separation distance between the two particles) fit more accurately to the actual path?

Thanks for your input.
 
How about Hermite Interpolation, which, according to Wikipedia, "allows us to consider given derivatives at data points, as well as the data points themselves. The interpolation will give a polynomial that has a degree less than or equal to the number of both data points and their derivatives, minus 1."

What do you think of this method? If someone knows of a source (book, article, etc.) where this method is explained in more detail than the Wikipedia article, I would greatly appreciate it if he/she let me know.

Thanks.
 
And to Hurkyl: I don't know how much it helps, but we are only dealing with a central force here (the "particles" are actually planets both moving along Keplerian orbits), so the particles won't be doing anything crazy in terms of their higher derivatives
 

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