[Proj] Ellipsoidal geodesic projections
Charles Karney
ckarney at sarnoff.com
Fri May 28 16:24:30 EST 2010
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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