How is a “conformal latitude” used in ellipsoidal stereographic projections?

At my work, I've been doing some maintenance on internal software that claims to implement a Stereographic projection whose formulas were copied from an old system a long time ago.

From what I've seen, it seems to implement the "Oblique and Equatorial Stereographic" from EPSG (code 9809). But there is one step that I was not able to identify. The "R" value used in our code is calculated with the following parameters:

(phi_n, lambda_n) - null distortion coordinates
(PHI_nc, LAMBDA_nc) - conformal null distortion coordinates
(PHI_o, LAMBDA_o) - conformal projection origin coordinates
e - eccenttricity
r_eq - earth radius at equator

Given this, the formula to get R is:

a = 1 + cos(PHI_nc) * cos(PHI_o) * cos(LAMBDA_nc - LAMBA_o)
b = sin(PHI_nc) * sin(PHI_o)
c = 2 * cos(PHI_nc) * [1 - (sin(phi_n) * e/2) ^ 2] ^ 1/2
R = r_eq * cos(phi_n) * (a + b) / c

Does anyone know what is going on here? I've been searching and reading about stereographic projections, but I wasn't able to find anything like that. In fact, I've not seen any projections that use this "null distortion coordinate."

Two things are happening here.

The first is the replacement of the actual latitude phi_n by the "conformal latitude" phi_nc. Think of this as distorting the ellipsoid (as specified by r_eq and e) into a perfect sphere. Because it is an ellipsoid of revolution, no change to the longitude occurs (lambda_n = lambda_nc), but the latitudes shift slightly. They do so in a way that is locally angle-preserving ("conformal").

The second is a slight adjustment of the scale of the stereographic projection, also to account for the shape of the ellipsoid. This is reflected in the variable c, which you can see depends on the eccentricity e (the sole determiner of the ellipsoid's shape).

Here is John Snyder's account:

...the ellipsoidal forms of the Stereographic projection are nonperspective, in order to preserve conformality. The oblique and equatorial aspects are also slightly nonazimuthal for the same reason. The formulas result from replacing geodetic latitude phi in the spherical equations with conformal latitude chi ... followed by a small adjustment to the scale at the center of projection.

Map Projections--A Working Manual, p. 160, emphasis added.

Incidentally, "null distortion coordinate" is an idiosyncratic term. According to Google, this thread is the only place on the Internet where such a phrase occurs!

• Thanks whuber. So, if I correctly understand, the factors that multiply r_eq are a math to calculate the scale factor (Ko)? The term "null distortion" is a translation (in the code I'm using, the variable names are in portuguese). Maybe somewhere else it does have a different name. – Alexandre Jun 29 '11 at 16:37
• @Alexandre (1) The correction is more than a constant scale factor: notice how it varies with latitude. (2) "Null distortion" is too vague, because variations on the same theme, such as authalic latitude, are used to control metric distortion in other kinds of projections of the ellipsoid. Therefore I doubt that the original Portuguese is a standard term, either. – whuber Jun 29 '11 at 16:49
• Also, in our code the R variable is called "conformal radius". – Alexandre Jun 29 '11 at 16:49
• After a long search, I've found the description of this specific projection. It's used by radar systems, and the document can be found on dtic.mil/cgi-bin/… – Alexandre Jul 13 '11 at 22:26
• @Alex Thanks. That document indeed uses the terms "conformal latitude" and "conformal longitude." – whuber Jul 14 '11 at 18:07