[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

(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

(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 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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