Yes, you're right, I made some approximations. The analysis is focused in the range of small \phi angles. However, it would be interesting to analyze what happe

Yes, you're right, I made some approximations. The analysis is focused in the range of small \phi angles. However, it would be interesting to analyze what happens when \phi gets big. To do so you'll need to solve -probably numerically- the Lagrange equation for \phi (it's the second equation in the post), and then plug the solution into the distance equation.