The Running Fix on an Ellipsoid

The running fix or sight-run-sight fix is a classic problem in celestial navigation. Methods employed to obtain the fix traditionally involve advancing a Line of Position (LoP) taken at an earlier time and crossing it with one obtained later. Attempts to generalise the operation of advancing an LoP when the Earth's surface is represented by a plane to the case of the sphere have resulted in proposals that contain poorly constrained approximations or are otherwise fundamentally flawed. A simple rapidly-convergent iterative procedure to obtain a running fix is described that avoids the notion of advanced LoPs and is readily applicable to both the sphere and ellipsoid.