[Proj] Ellipsoidal geodesic projections

[Mikael Rittri]

About ellipsoidal gnomonic, I had got the impression from 
your documentation of GeographicLib that it was possible
to construct such a projection with all geodesics exactly
straight.

> ...geodesic problems ... Bessel ... showed how the problem
> may be transferred to an "auxiliary sphere" where the latitude
> phi has been replaced by the reduced latitude beta where 
> tan(beta) = (1 - f)tan(phi). On this sphere, the geodesic
> is a great circle and the azimuth is the same as on the
> ellipsoid. However, the ellipsoidal distance is related to 
> the great circle distance by an integral; and the ellipsoidal
> longitude is similarly related to the longitude on the 
> auxiliary sphere. 
(http://geographiclib.sourceforge.net/html/geodesic.html) 

So I must have misunderstood this text. I guess "the geodesic" 
does not stand for an arbitrary geodesic but only one of 
those that intersect an origin point?

Mikael Rittri
Carmenta AB
Sweden
www.carmenta.com

-----Original Message-----
From: proj-bounces at lists.maptools.org
[mailto:proj-bounces at lists.maptools.org] On Behalf Of Charles Karney
Sent: Friday, May 28, 2010 11:25 PM
To: proj at lists.maptools.org
Subject: [Proj] Ellipsoidal geodesic projections

I've started playing with some projections that can be defined in terms
of ellipsoidal geodesics and would be interested to have references to
prior work on these or similar projections.  The ones I'm interested in
are:

(1) Oblique Cassini-Soldner: Like Cassini-Soldner but replace the origin
    meridian with a reference geodesic at an arbitrary azimuth.  (Thus,
    starting at origin proceed along reference geodesic v meters, turn
    clockwise 90deg and proceed along a geodesic u meters.)

(2) Two-point equidistant: Solve a triangle given 3 sides, described by
    Mauer (but only for a sphere??).  Easily generalizes to an
    ellipsoid.

(3) Two-point azimuthal: Solve a triangle given 2 angles and the
    included side, described by Mauer (but only for a sphere??).  Easily
    generalizes to an ellipsoid.  Geodesics are roughly straight.

(4) Gnomonic: For a sphere all geodesics are straight.  For an
ellipsoid,
    take the limit of the two-point azimuthal projection as the two
points
    approach one another.  All the geodesics intersecting the origion
are
    straight.  Nearby geodesics are approximately straight.

As far as I know, no-one has looked at (1), (2), or (3); but I'm
probably wrong about this.  Roy Williams, "Geometry of Navigation",
gives an ellipsoidal version of the gnomonic projection.  But this is
just a rescaling of the spherical version and does not keep the
geodesics very straight.

  --Charles

--
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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