[Proj] Spheroidal gnomonic projection

[Karney, Charles]

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