[PROJ] Ellipsoidal version of the gnomonic projection

[Charles Karney]

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.