Learn and optimize Andoyer distance

[x] Could we chop out some trigonometric or other functions to create a comparable andoyer?
[x] Learn Andoyer distance with Geographic Distance and Azimuth Calculations
We can not Calculate with The Area of a Sector of an Ellipse for:

On the surface of a sphere, geodesics lie along great circles (i.e. a circle with its center at the center of the sphere). An ellipsoid geodesic is a minimum-distance path on an ellipsoid, and you might be tempted to think it's a great-ellipse (an ellipse where the center of the ellipse is located at the center of the Earth). If you think it's a great-ellipse, you are right in some cases; in other cases, it is not an ellipse but an ill-defined oval of some sort.

http://nacc.upc.es/tierra/node16.html I can not open this link.
http://sci.tech-archive.net/Archive/sci.geo.satellite-nav/2004-12/2724.html I can not understand the formulas, do they write in LeTaX?
_http://home.att.net/~srschmitt/great_circle_route.html_ I can not open this link too.
http://www.codeguru.com/cpp/cpp/algorithms/article.php/c5115/Geographic-Distance-and-Azimuth-Calculations.htm I read this paper, but did not find the implement of andoyer distance, could you explain them for me? BTW I know the implement of ellipse model, could you please check the correctness?
https://futureboy.us/frinksamp/navigation.frink The first implementation of this page is vincenty, but the following function does not same as what andoyer_inverse.hpp wrote in Boost.Geometry, I did not figure out the formula of andoyer, so I tried to read these two codes to understand, but I failed, could you help me about this?
http://www.voidware.com/earthdist.html I can not open this link too.
http://www.dtic.mil/docs/citations/AD0627893
http://www.dtic.mil/docs/citations/AD703541 This two papers were so long, so I just browse them for a short time, do I need to learn them, or could you please help me to learn andoyer from them?

BoostGSoC17 / geometry