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.