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.
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