Curve fitting points in 3D space

Question

Trying to find functions that will assist us to draw a 3D line through a series of points.

For each point we know: Date&Time, Latitude, Longitude, Altitude, Spee

Answer 1

My guess is that a serious application of this would use a http://en.wikipedia.org/wiki/Kalman_filter. By the way, that probably wouldn't guarantee that the reported points were intersected either, unless you fiddled with the parameters a bit. It would expect some degree of error in each data point given to it, so where it thinks the object is at time T would not necessarily be where it was at time T. Of course, you could set the error distribution to say that you were absolutely sure you knew where it was at time T.

Short of using a Kalman filter, I would try and turn it into an optimisation problem. Work at the 1s granularity and write down equations like x_t' = x_t + (Vx_t + Vx_t')/2 + e,

Vx_t_reported = Vx_t + f,

Vx_t' = Vx_t + g where e, f, and g represent the noise. Then create a penalty function such as e^2 + f^2 + g^2 +... or some weighted version such as 1.5e^2 + 3f^2 + 2.6g^2 +... according to your idea of what the errors really are and how smooth you wnat the answer to be, and find the values that make the penalty function as small as possible - with least squares if the equations turn out nicely.