In spherical geometry, I need to know at what longitude λ a great circle arc φ1,λ1-φ2,λ2 has intersected a line of latitude φ.
I have found the equivalent equation for solving latitude φ for an unknown longitude λ on the excellent Williams Aviation Formulary, excerpted here:
It would seem all that is required is to rearrange this equation in terms of lon rather than lat, but performing the rearrangement is beyond me (and, it would seem, Wolfram Alpha).
λ = ?
Edit: Williams Aviation Formulary also explains 'Crossing Parallels' which seems to be what I need, but this finds the crossing points for a whole great circle rather than a great-circle arc. What is the most effective way of finding only the crossing of the arc?
Many thanks for any advise you can lend on this.
Best Answer
I would rewrite the first equation as $\tan \phi = ...$ Then manipulate it to get an expression of the form $k = a \sin(\lambda-c) + b \sin (\lambda-d)$ where $a$, $b$, $c$, $d$ and $k$ do not depend on $\lambda$.
Wolfram Alpha provides solutions to the latter equation.