How to use linear interpolation estimate current position between two Geo Coordinates?

Question

I have the following available:

• last reported lat,lon w/timestamp
• target lat,lon
• estimated time to target
• heading

How

Answer 1

You want to use a Slerp, or spherical linear interpolation.

Convert your latitude and longitude to a unit 3-vector:

p=(x,y,z)=(cos(lon)*cos(lat), sin(lon)*cos(lat), sin(lat))

Then, "Slerp" gives you a constant-velocity interpolation along the surface of the unit sphere:

theta= angle between 3-vectors p0 and p1 (e.g., cos(theta)= p0.p1)
Slerp(p0,p1,t)= ( p0*sin((1-t)*theta) + p1*sin(t*theta) ) / sin(theta)

Note that if theta is very close to 0 or 180 degrees, this formula can be numerically unstable. In the small-angle case, you can fall back to linear interpolation; in the 180 degree case, your path is genuinely ambiguous.