# [Proj] Ellipsoidal Orthographic

Noel Zinn (cc) ndzinn at comcast.net
Tue Jun 28 15:25:32 EST 2011

```Thanks, Charles.  I'd like to see that.  You have my e-dress ... or post to
the list.  -Noel

Noel Zinn, Principal, Hydrometronics LLC
+1-832-539-1472 (office), +1-281-221-0051 (cell)
noel.zinn at hydrometronics.com (email)
http://www.hydrometronics.com (website)

-----Original Message-----
From: Charles Karney
Sent: Tuesday, June 28, 2011 12:59 PM
To: PROJ.4 and general Projections Discussions
Cc: Noel Zinn (cc)
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:
>