Interpolation

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BOCCARA Michael (MICHAEL.BOCCARA++at++siege.aerospatiale.fr)
Tue, 28 Oct 1997 19:57:30 +0100

  Thanks very much to Frederic Francis, Scott McMillan, Jason M. Coposky, and
Simon Mills, for their answers about Kinematics Interpolation.
Meanwhile, I found the formula for computing a cubic spline between 2
pos-velocity points , and in a good will, I give it to you :

Be M1, V1, M2, V2, the position and velocity vectors at times t1 and t2.
Be t a time between t1 and t2

the segment index for time t :
   u = (t - t1) / (t2 - t1);
Spline vector polynom for the (t1, t2) segment
    A = (V1 + V2) * DT - 2 * (M2 - M1);
    B = - (2 * V1 + V2) * DT + 3 * (M2 - M1);
    C = V1 * DT;
    D = M1;
(DT = t2 - t1)
Interpolated position at time t :
    M = A * u * u * u + B * u * u + C * u + D;
  
So smooth !

Those results can easily be found by derivating the above formula, applying
the t1 and t2 edge constraints, and inverting a 4x4 matrix.

Hope it will help the interested people, among whom the kind ones who did
answer me.

Mike

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