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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.