At 02:43 PM 6/12/02 -0500, Clifford J Mugnier wrote: >Computing Grid Convergence angle (grivation) at a point with the "brute >force method" (suggested below) is valid only for short lines. For lines >over 5 miles in length, one must use the (T-t) correction. Alternatively, >compute the direct solution of the ellipsoidal geodesic. > >For the formulae accessable thru the internet, go to www.NIMA.mil and >download TR8358.2 pages 2-1 thru 2-7.
Many thanks to Dr. Mugnier for those thoughts and that Web reference. That site does not seem to have a link to the Technical Manual, but it is freely available as a 3 MB pdf at http://164.214.2.59/GandG/puborder.html . In my experience Mugnier is the final authority on issues like this, but some comments on this calculation might be helpful to the intrepid reader who really intends to compute grid convergence. When you get down to it, the "brute force method" is far, far simpler to implement in a GIS environment and is likely at least as accurate as the formulae Mugnier cites--and maybe even more accurate. There is no problem with "lines over 5 miles in length" with this simple method, because it uses lines of only "a few meters" (see the quote below). Its calculation requires projecting two nearby points and performing some simple trigonometry with the results. There is potential loss of numerical precision if one is too aggressive and uses an extremely small distance, but by being judicious it is not difficult to compute grid convergences to single precision at least. I have had good success using distances between one and one hundred meters; these recommended values are supported by estimates of numerical precision, but I will spare you the math. The direct solution in TR 8358.2 is a truncated power series in delta-lambda (longitude relative to the central meridian) whose coefficients are themselves complicated polynomials involving the eccentricity and trigonometric functions of the latitude: in short, they're not any nicer to code than performing the projections from scratch. So, if you are working in a GIS, the GIS in all likelihood already provides the projection capability for you and you ought to avoid the formulas in TR 8358.2. If you are in a different programming environment and have to do things from scratch, and all you need is the grid convergence, then implementing the TR 8358.2 formulas makes sense. But as a matter of software engineering, you might be happier just implementing the projection formulas from scratch and then using the "brute force" method to obtain the convergence, because it's much easier to find data to check the accuracy of your program. Dr. Mugnier has suggested in past postings on other lists, if I have understood him correctly, that certain GISes do not implement their projections very accurately. The nice thing about the finite-difference approach is that any systematic local error, such as might be introduced by truncating a power series too early, just about cancels out in the finite-difference computation (especially when you compute centered differences), so you can still get very high-accuracy results. That really was the point of the original post: you might not have realized it, but as soon as you have high-precision projection capabilities, you can derive a lot of interesting values almost for free by using finite-difference approximations to the derivatives. These values include not only grid convergence, but scale factors, axes and orientation of Tissot's indicatrix, principal curvatures, and any other local properties of the projection and the surface that you like. --Bill Huber >>These properties are defined in >>terms of partial derivatives of the projection, which in all practical >>cases are extremely well approximated by finite differences computed over >>distances of a few meters. Thus, for the convergence, simply create a >>short north-pointing geodesic centered at the point in question and project >>it. --------------------------------------------------------------------- List hosting provided by Directions Magazine | www.directionsmag.com | To unsubscribe, e-mail: [EMAIL PROTECTED] For additional commands, e-mail: [EMAIL PROTECTED]
