# Finding a point along the surface of a ellipsoid

I have read the answers for both of these questions -Finding a point along a line a certain distance from another point and Find a point along line on Earth and I am trying to convert the transformations given in this question that I asked here - Trigonometric formula derivation into their spherical trigonometric versions. All longitudes and latitudes have been defined with respect to the WGS84 datum

$P(\lambda^,\phi^)$ lies on a geodesic curve from $O(\lambda_0,\phi_0)$ and I want the coordinates of $R(\lambda,\phi)$ which presumably lies along the geodesic distance between P and O. So O is what is called the nadir point of the satellite on the earth's surface. P is the coordinates of a ray at a slant angle $\theta$ from the location of the satellite above the earth's surface.

I have also been the height and from that I am presuming the arc length PR can be deduced by the following equation $$h * tan(\theta)$$ and this arc lies along the geodesic OP.

I am wanting to find out the coordinates of $R(\lambda,\phi)$ where R is the coordinates of the actual cloud top position but I have been given the coordinates of P which is the apparent cloud top position. Hence a parallax correction needs to be applied.

I believe the Matlab/Octave function reckon does what I want but I want to do this in my own programming language and hence I cannot use that API call. I would like to do this from first principles.

• See also physics.stackexchange.com/questions/25917/… (though there is no answer there right now). – davidlowryduda Apr 26 '18 at 11:24
• @mixedmath - yes I flagged that for the moderators to look at. – gansub Apr 26 '18 at 11:44
• Yes, that's how I became aware of it. You'd asked whether these should be merged or closed as duplicates (neither of which are possible since they're on different sites). But what my comment does is let others who visit this question know that there is another place which might have an answer, and thus reduce duplication of effort. – davidlowryduda Apr 26 '18 at 13:07