[Proj] Spheroidal gnomonic projection
Karney, Charles
ckarney at Sarnoff.com
Sun Jun 13 21:49:10 EST 2010
I've tracked down another ellipsoidal gnomonic projection
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).
As described by Bugayevskiy and Snyder, this consists of a conformal
projection to a sphere followed by a central projection onto a tangent
plane. However, on reading the paper I see that there's an important
wrinkle: the conformal projection is offset in latitude in order to make
the geographic and conformal latitudes match at the center point
(instead of the equator), i.e.,
asinh(tan(beta)) = asinh(tan(phi))
- e * ( atanh( e * sin(phi) ) - atanh( e * sin(phi0) ) )
With this projection, normal sections are mapped to straight lines. The
maximum error in the direction of a geodesic within a radius r of the
center point scales as f*(r/a)^2.
In summary, we have 3 ellipsoidal gnomonic projections
* center projection, Williams (1998)
great ellipses are straight
error in azimuth for geodesics = f*(r/a) = 108" for r = 1000km
* conformal mapping, Letoval'tsev (1963)
normal sections through center point are straight
error in azimuth for geodesics = f*(r/a)^2 = 17.4" for r = 1000km
* geodesic mapping, Karney (2010)
geodesics through center point are straight
error in azimuth for geodesics = 0.4*f*(r/a)^3 = 1.04" for r = 1000km
Postscript: The affiliation of Letoval'tsev is listed as "Moscow
Institute of Railroad Transportation Engineers". I wonder how often you
see "Railroad" appearing in the addresses of articles on geodesy
nowadays.
--
Charles Karney <ckarney at sarnoff.com>
Sarnoff Corporation, Princeton, NJ 08543-5300
Tel: +1 609 734 2312
Fax: +1 609 734 2662
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