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<p style="line-height: 100%; margin-bottom: 0in">
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. </p>
<p style="line-height: 100%; margin-bottom: 0in">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. </p>
<p style="line-height: 100%; margin-bottom: 0in">Noel</p>
<p></p>
<div class="moz-cite-prefix">On 12/28/2022 6:18 PM, Charles Karney
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:5468a9aa-31b0-6e52-e8d9-fd6f5d1eb7ff@karney.com">Noel,
<br>
<br>
The projection you describe, a central projection the ellipsoid
onto a
<br>
plane, Method 1, is the ellipsoidal generalization of the gnomonic
<br>
projection suggested in 2 papers from 1997:
<br>
<br>
 B. R. Bowring,
<br>
 The central projection of the spheroid and surface lines
<br>
 <a class="moz-txt-link-freetext" href="https://doi.org/10.1179/sre.1997.34.265.163">https://doi.org/10.1179/sre.1997.34.265.163</a>
<br>
<br>
 R. Williams,
<br>
 Gnomonic projection of the surface of an ellipsoid
<br>
 <a class="moz-txt-link-freetext" href="https://doi.org/10.1017/S0373463300023936">https://doi.org/10.1017/S0373463300023936</a>
<br>
<br>
Much earlier, Letoval'tsev proposed a different generalization,
Method
<br>
2:
<br>
<br>
 I. G. Letoval'tsev,
<br>
 Generalization of the gnomonic projection for a spheroid and the
<br>
 principal geodetic problems involved in the alignment of surface
<br>
 routes,
<br>
 Geodesy and Aerophotography, 5, 271-274 (1963),
<br>
 translation of Geodeziya i Aerofotos'emka 5, 61-68 (1963).
<br>
<br>
Let's call my method (the limit of a double azimuthal projection),
<br>
Method 3.
<br>
<br>
In my 2013 paper, I compared all three methods finding the maximum
<br>
deviation, h, of straight line segments in the gnomonic projection
from
<br>
the geodesic where the endpoints lie within a radius r of the
center of
<br>
projection. I found that h/r scaled as
<br>
<br>
  r/a   for Method 1
<br>
 (r/a)^2 for Method 2
<br>
 (r/a)^3 for Method 3
<br>
<br>
where a is the equatorial radius of the ellipsoid. If the
straightness
<br>
of geodesics is the desired property of the projection (as it is
for
<br>
seismic work and for radio direction finding), then Method 3 (the
method
<br>
implemented in the proposed PR) is the best choice.
<br>
<br>
On 12/28/22 17:30, Noel Zinn wrote:
<br>
<blockquote type="cite">The following is a link to a derivation of
an ellipsoidal gnomonic via ECEF that I did 12 years ago:
<br>
<br>
<a class="moz-txt-link-freetext" href="http://www.hydrometronics.com/downloads/Ellipsoidal%20Gnomonic%20Projection.pdf">http://www.hydrometronics.com/downloads/Ellipsoidal%20Gnomonic%20Projection.pdf</a>
<br>
<br>
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 linesâ€. 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 planeâ€, the same
method of derivation of the spherical version?
<br>
<br>
Noel
<br>
<br>
On 12/28/2022 3:59 PM, Clifford J Mugnier wrote:
<br>
<blockquote type="cite">The spherical gnomonic projection has
the property that geodesics map to
<br>
> straight lines.
<br>
</blockquote>
</blockquote>
</blockquote>
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