# Interpolating a set of GPS points on ellipsoid earth model.

I have a set of GPS (latitude, longitude) co-ordinates along with the time at which the coordinates were collected. Additionally, I have the speed and the heading of the vehicle at those coordinates. I want to find an interpolating function so that I can determine the position (lat, long), speed, and heading of the vehicle at a time between the tabulated points. (My "vehicle" could be any land, marine, or air vehicle. No need to worry about routes. I'm simply looking for shortest-distance interpolation.)

I have to use the ellipsoid earth model (WGS84), and I can transform any given latitude and longitude to distance and heading in the desired reference frame.

More precisely, my tabulated data is as follows:

$$F = \lbrace{ x_{i}, y_{i}, s_{i}, h_{i}, t_{i}; 0 \leq i \leq N-1; i\in \mathbb{Z} \rbrace}$$ where $x_{i}$ is the latitude, $y_{i}$ is the longitude, $s_{i}$ is the speed, $h_{i}$ is the heading, and $t_{i}$ is the time. The longitude and latitude can be transformed into distance and heading in a desired reference frame.

My questions are as follows:

1) When computing the interpolating function for position, am I correct in assuming that this can only be solved using a 2-dimensional interpolation method (I would have to interpolate the distance and heading for the desired point in time separately).

2) Is there any method of interpolation that is able to use the derivative at discrete points to provide a better approximation of the function? For my problem, I have the position and the derivative (speed and heading) at discrete points. I'm familiar with different methods of interpolation (polynomial, cubic spline, b-spline, and e.t.c), but none of these methods use the derivative to provide a better approximation. So I'm left with computing the interpolating function for the position, speed, and heading separately.

• Given the definition of the problem it seems that unless your "vehicle" is a ship or an airplane the best possible interpolation will probably use GIS data from the given area to guess the most probbable route. Of course this assumes that such an approach is feasible. – DRF Jun 9 '15 at 14:30
• Any other approximation seems likely to not really gain much information from the speed and little from the heading if the vehicle is an actual vehicle since the speed and the heading at a point has little to do with what is going to be really happening to the vehicle (traffic, roads etc. will have much more influence). This would change assuming the points are quite close together (say every couple of seconds) in which case a spline interpolation should be able to use speed and heading together (taken as the derivative vector) if you use a large enough polynomial to be able to use it. – DRF Jun 9 '15 at 14:35
• Yes, assume that my "vehicle" is a ship or an airplane. No need to worry about routes; I just want to interpolate over curved geometry. – MRashid Jun 9 '15 at 14:37