How to interpolate ECEF coordinates on an WGS84 ellipsoid - coordinates

Question

Is there a direct method (not involving converting the coordinates to lat/lon) to interpolate between 2 ECEF coordinates (xyz) in order for the interpolated point to be located on the WGS84 ellispoid. The original 2 points are computed from geodetic coordinates.
Interpolating on a sphere seem obvious but I can't seem to derive a solution for the ellipsoid.
Thank you in advance.

Answer 1

Let assume you got 2 points p0(x,y,z) and p1(x,y,z) and want to interpolate some p(t) where t=<0.0,1.0> between the two.
you can:
rescale your ellipsoid to sphere
simply like this:
const double mz=6378137.00000/6356752.31414; // [m] equatoreal/polar radius of Earth
p0.z*=mz;
p1.z*=mz;
now you got Cartesian coordinates refering to spherical Earth model.
interpolate
simple linear interpolation would do
p(t) = p0+(p1-p0)*t
but of coarse you also need to normalize to earth curvature so:
r0 = |p0|
r1 = |p1|
p(t) = p0+(p1-p0)*t
r(t) = r0+(r1-r0)*t
p(t)*=r/|p(t)|
where |p0| means length of vector p0.
rescale back to ellipsoid
by dividing with the same value
p(t).z/=mz
This is simple and cheap but the interpolated path will not have linear time scale.
Here C++ example:
void XYZ_interpolate(double *pt,double *p0,double *p1,double t)
{
const double mz=6378137.00000/6356752.31414;
const double _mz=6356752.31414/6378137.00000;
double p[3],r,r0,r1;
// compute spherical radiuses of input points
r0=sqrt((p0[0]*p0[0])+(p0[1]*p0[1])+(p0[2]*p0[2]*mz*mz));
r1=sqrt((p1[0]*p1[0])+(p1[1]*p1[1])+(p1[2]*p1[2]*mz*mz));
// linear interpolation
r = r0 +(r1 -r0 )*t;
p[0]= p0[0]+(p1[0]-p0[0])*t;
p[1]= p0[1]+(p1[1]-p0[1])*t;
p[2]=(p0[2]+(p1[2]-p0[2])*t)*mz;
// correct radius and rescale back
r/=sqrt((p[0]*p[0])+(p[1]*p[1])+(p[2]*p[2]));
pt[0]=p[0]*r;
pt[1]=p[1]*r;
pt[2]=p[2]*r*_mz;
}
And preview:
Yellow squares are the used p0,p1 Cartesian coordinates, the White curve is the interpolated path where t=<0.0,1.0> ...