# Calculate accelerations at a given point from accelerations at 3 other points on a rigid body

I've measured accelerations on a vibrating rigid body using 3 accelerometers which measure in 3 directions. So I've 3 vectors with three entries measuring accelerations over time at 3 points (1,2,3):

a1(t)={ẍ1(t),ÿ1(t),z̈1(t)},
a2(t)={ẍ2(t),ÿ2(t),z̈2(t)},
a3(t)={ẍ3(t),ÿ3(t),z̈3(t)}


Further I have the coordinates of those three points with regard to the rigid body. Now I would like to obtain the resulting accelerations at a fourth point on the rigid body for which I also have the coordinates. What would be a suitable algorithm for that?

So far I have found http://graphics.stanford.edu/~smr/ICP/comparison/eggert_comparison_mva97.pdf and Given 3 points of a rigid body in space, how do I find the corresponding orientation (aka rotation or attitude)? however I am dealing with accelerations and not with deflections. The sensors only measure above 1 Hz, so it isn't possible to derive the deflections from the accelerations. So I don't have two sets of coordinates for the 3 points at two timestamps (t0 and t1) but I have three points and two sets of acceleration vectors at multiple timestamps. With this information, how do I get the acceleration vector at a fourth point over time?

Further I wonder whether it is possible to calculate the angular accelerations of the body (e.g. 3 translations, 3 rotations over time).