[PROJ] Ellipsoidal version of the gnomonic projection

[Charles Karney]

Thanks for the feedback, Noel,

Users who need a projection that maps great ellipses to straight lines
can use the gnomonic projection on a sphere (i.e., merely use +f=0).
This is equivalent to a central projection for the ellipsoid with a
stretch in one direction (which preserves straight lines, of course).

Incidentally one nice property preserved by the double azimuthal
generalization of the gnomonic projection is that it is conformal near
the center point.  This is not true with the central projection of an
ellipsoid.

   --Charles

On 12/28/22 21:11, Noel Zinn wrote:
> Thank you, Charles, for the links to my predecessors (Bowring and 
> Williams). Their papers are expensive from Taylor & Francis. Since 
> retirement my spare change has gone into photographic gear and not 
> cartography papers! But I do notice from Bowring’s (free) abstract that 
> Method 1 straight lines are great elliptic lines on the ellipsoid, 
> which, according to Wikipedia differ “within one part in 500,000” from 
> the geodesic distance, not too shabby. But no free insight into 
> azimuthal difference, which is your primary concern. Anyway, Method 1 
> does provide something (straight elliptic lines as a substitute for 
> straight geodesics) and it’s derived similarly to the spherical 
> gnomonic. That’s appealing, to me at least.
> 
> OTOH, your concern is to approximate a property of the spherical 
> gnomonic that doesn’t exist exactly in the ellipsoidal case. Your 
> evidence of (r/a)^3 for Method 3 for azimuthal discrepancy is 
> compelling; that’s a very small number. My (original) copy of your 2013 
> paper is at Texas A&M today, but I see that Springer offers it for free. 
> Many thanks for that … and all you (and so many others) do for Proj by 
> the way. Without rereading Section 8 (yet) your comment that Method 3 
> degenerates to the spherical gnomonic is reassuring to me. I hope I read 
> that correctly. No doubt that Method 3 is an improvement over the status 
> quo.
> 
> Noel
> 
> On 12/28/2022 6:18 PM, Charles Karney wrote:
>> Noel,
>>
>> The projection you describe, a central projection the ellipsoid onto a
>> plane, Method 1, is the ellipsoidal generalization of the gnomonic
>> projection suggested in 2 papers from 1997:
>>
>>   B. R. Bowring,
>>   The central projection of the spheroid and surface lines
>> https://doi.org/10.1179/sre.1997.34.265.163
>>
>>   R. Williams,
>>   Gnomonic projection of the surface of an ellipsoid
>> https://doi.org/10.1017/S0373463300023936
>>
>> Much earlier, Letoval'tsev proposed a different generalization, Method
>> 2:
>>
>>   I. G. Letoval'tsev,
>>   Generalization of the gnomonic projection for a spheroid and the
>>   principal geodetic problems involved in the alignment of surface
>>   routes,
>>   Geodesy and Aerophotography, 5, 271-274 (1963),
>>   translation of Geodeziya i Aerofotos'emka 5, 61-68 (1963).
>>
>> Let's call my method (the limit of a double azimuthal projection),
>> Method 3.
>>
>> In my 2013 paper, I compared all three methods finding the maximum
>> deviation, h, of straight line segments in the gnomonic projection from
>> the geodesic where the endpoints lie within a radius r of the center of
>> projection.  I found that h/r scaled as
>>
>>    r/a    for Method 1
>>   (r/a)^2 for Method 2
>>   (r/a)^3 for Method 3
>>
>> where a is the equatorial radius of the ellipsoid.  If the straightness
>> of geodesics is the desired property of the projection (as it is for
>> seismic work and for radio direction finding), then Method 3 (the method
>> implemented in the proposed PR) is the best choice.
>>
>> On 12/28/22 17:30, Noel Zinn wrote:
>>> The following is a link to a derivation of an ellipsoidal gnomonic 
>>> via ECEF that I did 12 years ago:
>>>
>>> http://www.hydrometronics.com/downloads/Ellipsoidal%20Gnomonic%20Projection.pdf
>>>
>>> Being thoroughly retired now, and having donated my entire technical 
>>> library to a university, I’m not in a position to test this 
>>> ellipsoidal version against the criteria stated by Charles, namely 
>>> “The spherical gnomonic projection has the property that geodesics 
>>> map to straight lines”. That may, or may not, be true for this 
>>> ellipsoidal version. But does that matter if this “ellipsoidal 
>>> gnomonic is a direct perspective from the geocenter through the 
>>> ellipsoid onto the tangential plane”, the same method of derivation 
>>> of the spherical version?
>>>
>>> Noel
>>>
>>> On 12/28/2022 3:59 PM, Clifford J Mugnier wrote:
>>>> The spherical gnomonic projection has the property that geodesics 
>>>> map to
>>>> > straight lines.