[Proj] Re: Distance calculations

strebe at aol.com strebe at aol.com
Thu Oct 16 14:24:29 EDT 2008

In solving the second problem of geodesy (= the inverse problem = geodesic segment length and azimuth given two points) in the most general cases, one must determine whether the two endpoints straddle a "node" or not. (A node is the highest or lowest latitude a complete geodesic reaches along its travel.) Whether it straddles a node or nodes determines how one sets up the limits of integration (and indeed how many instances of the integral come into play.) Since you have no information about the geodesic a priori, this can be quite difficult. Computations of elliptic integrals of the third kind also come into play, and, in the general case, these can be problematic. Bulirsch's algorithm, for example, does not come with the same accuracy guarantees of his solutions for elliptic integrals of the first and second kinds, and is known to deteriorate with certain ranges of parameters.

Helmert formulated an iterative solution to the second problem which, as far as I know, is the basis of all published solutions. In pursuit of computational efficiency, the published solutions truncate series, rearrange terms, fold together steps, and otherwise cut corners. In the absence of rigorous analysis, there is no guarantee that these operations were even correctly formulated, let alone that they carry "sufficient" accuracy. Since benchmarks are so few and come themselves with unknown errors, the question of accuracy is hardly just academic.

Lastly, one question I've never seen addressed is the suitability and accuracy of any of these methods for spheroids that are not earthlike. If we have not a single published solution to the second problem that we can apply to ellipsoids of (relatively) high eccentricity, are we not restricted in planetary research?

-- daan Strebe


-----Original Message-----
From: Christopher Barker <Chris.Barker at noaa.gov>
To: PROJ.4 and general Projections Discussions <proj at lists.maptools.org>
Sent: Thu, 16 Oct 2008 9:04 am
Subject: Re: [Proj] Distance calculations

I've got a question:?

I understand how these issues can be academically interesting, but is 
there any practical difference??

Surely the ellipsoidal model of the earth is an approximation anyway -- 
how accurate is it, and are the difference between the various 
algorithms under discussion really larger than the error inherent in the 
ellipsoidal approximation??

Just wondering, it's the engineer in me.?


Christopher Barker, Ph.D.?


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