[Proj] Ellipsoidal Orthographic

[Charles Karney]

Cliff,

My remarks were about the orthographic projection; so in this case we
are talking about a point on the surface of the ellipsoid.

However, the closed form solution of the 3d problem (given [x,y,z], find
[phi,lam,h]) which entails solving a quartic equation *can* be made
accurate for any values of [x,y,z].  If you want to give me some
"challenge" coordinates to test out, I can feed them into
GeographicLib's CartConvert to see what pops out.

   --Charles

On 06/28/11 14:41, Clifford J Mugnier wrote:
> Charles,
> 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.
> Cliff
> Clifford J. Mugnier, C.P., C.M.S.
> Chief of Geodesy,
> *Center for GeoInformatics*
> Department of Civil Engineering
> Patrick F. Taylor Hall 3531
> *LOUISIANA STATE UNIVERSITY *
> Baton Rouge, LA70803
> 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
>
> Noel,
>
> 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
> expected.
>
> I can provide you with the outline of the solution if you want.
>
> --Charles
>
>
> 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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-- 
Charles Karney <charles.karney at sri.com>
SRI International, Princeton, NJ 08543-5300
Tel: +1 609 734 2312