[PROJ] Ellipsoidal version of the gnomonic projection

Wed Dec 28 15:44:16 PST 2022

Clifford: point taken.  But would this comment apply to the gnomonic
projection?  Historically this has been a spherical-only projection and
so extending it (in some reasonable way) to an ellipsoid won't tread on
anyone's toes.

   --Charles

On 12/28/22 16:59, Clifford J Mugnier wrote:
> Mathematically correct is not necessarily LEGALLY correct.  For instance 
> with regard to France and to many French colonies, the French Army 
> Lambert Truncated Cubic Conic is only partially conformal, but is a 
> perfectly LEGAL projection for computing cadastral boundaries BEFORE 
> 1948.  After 1948, the French Legislature changed the Law to require the 
> perfectly conformal mathematical equations for France, and many French 
> colonies followed suit - except for Algeria.
> 
> Mathematical elegance is nice but not always correct under the Law for a 
> particular place.  The EPSG tries to pay attention to that; I am not so 
> sure about ESRI.
> 
> Same goes for the many various truncations of the Transverse Mercator. 
> It depends on _where and when for what equations._
> 
> Clifford J. Mugnier, c.p., c.m.s.
> 
> Chief of Geodesy, /(Emeritus)/
> 
> LSU Center for GeoInformatics (ERAD 266)
> 
> Dept. of Civil Engineering
> 
> *LOUISIANA STATE UNIVERSITY*
> 
> Baton Rouge, LA  70803
> 
> Research:   (225) 578-4578
> 
> Cell:           (225) 328-8975
> 
> honorary lifetime member, lsps
> 
> fellow emeritus, asprs
> 
> member, apsg
> 
> ------------------------------------------------------------------------
> *From:* PROJ <proj-bounces at lists.osgeo.org> on behalf of Even Rouault 
> <even.rouault at spatialys.com>
> *Sent:* Wednesday, December 28, 2022 3:43 PM
> *To:* charles at karney.com <charles at karney.com>; proj <PROJ at lists.osgeo.org>
> *Subject:* Re: [PROJ] Ellipsoidal version of the gnomonic projection
> I'm favorable to this improvement. This isn't the first time we add an
> ellipsoidal formulation to a projection method that had only a spherical
> one (last time was for orthographic). As I mentioned in the PR, it would
> be great if we know how to deal with CRS definitions under the Esri
> authority that use gnomonic to hopefully be consistent with what they
> do, at least on CRS under their authority, as Esri has several flavors
> of gnomonic. Hopefully we'll get some input from Esri people. I'll try
> to reach with them.
> 
> Even
> 
> Le 28/12/2022 à 22:01, Charles Karney a écrit :
>> I've submitted a pull request (PR)
>>
>>   https://nam04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgithub.com%2FOSGeo%2FPROJ%2Fpull%2F3522&data=05%7C01%7Ccjmce%40lsu.edu%7Ccd3df9623f73492cd90808dae91c9044%7C2d4dad3f50ae47d983a09ae2b1f466f8%7C0%7C0%7C638078606138115575%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=KQ9%2FVZ4BQb1XKztml3zCOeYSWMlK%2FyAfY0NaIkyCWHU%3D&reserved=0 <https://nam04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgithub.com%2FOSGeo%2FPROJ%2Fpull%2F3522&data=05%7C01%7Ccjmce%40lsu.edu%7Ccd3df9623f73492cd90808dae91c9044%7C2d4dad3f50ae47d983a09ae2b1f466f8%7C0%7C0%7C638078606138115575%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=KQ9%2FVZ4BQb1XKztml3zCOeYSWMlK%2FyAfY0NaIkyCWHU%3D&reserved=0>
>>
>> for PROJ to implement an ellipsoidal generalization of the gnomonic
>> projection.
>>
>> The spherical gnomonic projection has the property that geodesics map to
>> straight lines.
>>
>> There's no projection with the same property for the ellipsoid.
>> However, the spherical gnomonic projection is also the limit of a double
>> azimuthal projection, a projection which preserves the azimuths from two
>> points, in the limit that the two points coalesce.  This property
>> carries over to the ellipsoidal version of the projection implemented by
>> this PR.  For earth ellipsoids, geodesics which pass within a few
>> hundred km of the center of the projection are very nearly straight.
>> For details and for a derivation of the projection, see Sec. 8 of my
>> 2013 paper
>>
>>   Algorithms for geodesics
>>   https://nam04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1007%2Fs00190-012-0578-z&data=05%7C01%7Ccjmce%40lsu.edu%7Ccd3df9623f73492cd90808dae91c9044%7C2d4dad3f50ae47d983a09ae2b1f466f8%7C0%7C0%7C638078606138115575%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=D1Q2L9O6vA9mRVwlzX%2FWzmhfdLl5RHuIMU%2FdpU%2Fjels%3D&reserved=0 <https://nam04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1007%2Fs00190-012-0578-z&data=05%7C01%7Ccjmce%40lsu.edu%7Ccd3df9623f73492cd90808dae91c9044%7C2d4dad3f50ae47d983a09ae2b1f466f8%7C0%7C0%7C638078606138115575%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=D1Q2L9O6vA9mRVwlzX%2FWzmhfdLl5RHuIMU%2FdpU%2Fjels%3D&reserved=0>
>>
>> Merging this PR will result in a minor incompatibility:  With the
>> gnomonic projection in current version of PROJ, an ellipsoid is treated
>> (inconsistently!) as a sphere whose radius matches the equatorial radius
>> of the ellipsoid.  After the merge, the ellipsoidal gnomonic projection
>> will be used (and you will get somewhat different results).   To retain
>> the previous behavior you'll need to set the flattening of the ellipsoid
>> to zero, +f=0.
>>
>> Comments are welcome...
>>
> -- 
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> My software is free, but my time generally not.
> 
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-- 
Charles Karney <charles at karney.com>
702 Prospect Ave
Princeton, NJ 08540-4037