[Proj] Ellipsoidal Orthographic

Clifford J Mugnier cjmce at lsu.edu
Tue Jun 28 13:41:28 EST 2011

Using the closed form for the geocentric inverse is OK for surface mapping, but for a more universal application that will work with aerial photography and particularly with satellite imagery, the closed form deteriorates with altitude above the ellipsoid.  The iterative form will accommodate orbital imagery altitudes with no loss in computational precision.  That's a significant consideration since the "Oil Patch" indeed uses orbital imagery in their exploration activities.
Clifford J. Mugnier, C.P., C.M.S.
Chief of Geodesy,
Center for GeoInformatics
Department of Civil Engineering 
Patrick F. Taylor Hall 3531
Baton Rouge, LA  70803
Voice and Facsimile:  (225) 578-8536 [Academic] 
Voice and Facsimile:  (225) 578-4578 [Research] 
Cell: (225) 328-8975 [Academic & Research]
Honorary Life Member of the 
Louisiana Society of Professional Surveyors 
Fellow Emeritus of the ASPRS 
Member of the Americas Petroleum Survey Group


From: proj-bounces at lists.maptools.org on behalf of Charles Karney
Sent: Tue 28-Jun-11 12:59
To: PROJ.4 and general Projections Discussions
Subject: Re: [Proj] Ellipsoidal Orthographic


You use an iterative method to solve for the reverse projection.  But
the reverse projection can be written in closed form.  Recovering the 3d
position [x,y,z] from the easting and northing entails solving a
quadratic equation, so that you get 2 or 0 (or exceptionally 1) root, as

I can provide you with the outline of the solution if you want.


On 06/28/11 07:43, Noel Zinn (cc) wrote:
> The only equations for the ellipsoidal orthographic that I've ever found
> published (in a book or journal) are those of Bugayevskiy and Snyder (1995),
> which are complicated and (the authors acknowledge) truncated.  Following
> EPSG Guidance Note 7, Part 2, I've prepared a presentation on the
> ellipsoidal orthographic that offers simple, exact equations.  The
> derivation also suggests that the ellipsoidal orthographic is unique among
> projections, being transitional between distorted cartography in 2D and
> undistorted visualization in 3D on a computer in ECEF or ENU (topocentric)
> coordinates.  A link to the presentation follows:
> http://www.hydrometronics.com/downloads/Ellipsoidal%20Orthographic%20Projection.pdf
> Does anyone in this group have other sources of information on the
> ellipsoidal orthographic?
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