# Non-iterative algorithms for direct problem in the study of geodesic

I am finding an appropriate algorithm to implement in my program for direct problem in the study of geodesic. In brief, the direct problem deals with finding the end point (its longitude, latitude and azimuth) given the start point (its longitude, latitude and azimuth) and the distance from the start point to end point.

At the moment, I found out that Vincenty's formulae is a well-known algorithm for solving the geodesic direct problem. It is an iterative algorithm. According to this post, GeographicLib also has its own algorithms to solve the direct problem.

I have two questions. The first one is whether there are any non-iterative algorithms for this problem? If yes, can you give the name of the method or reference to it. The second one is if it is possible, can you give me the information about the algorithms used by GeographicLib.

• Welcome to GIS SE. As a new user, please take the Tour, which emphasizes the importance of asking one question per Question. The Direct problem is a partial differential equation which is only solvable through iterative means. As GeographicLib is open source, you should review the code before asking about the algorithm. Any questions about the code might be better asked of the source maintainers, though if you include the relevant code block here with a link to the rest, we might be able to answer. – Vince Jul 4 '18 at 12:06

## 1 Answer

The algorithms used by GeographicLib are given in my paper, Algorithms for geodesics. The solution of the direct problem presented there is non-iterative. This is accomplished by reverting the series for the distance in terms of the angle on the auxiliary sphere. This idea originates with Oriani (1833).

• Thank you! I will read your paper to understand about the algorithm. – N.Hung Jul 5 '18 at 1:05
• I found a solution on stackoverflow but I'm not satisfied with it. This solution is a non-iterative one, but it looks like an approximation. Can you give some comments on this method. – N.Hung Jul 5 '18 at 1:18
• Yes, this is an approximate method. – cffk Jul 5 '18 at 23:43