StereographicProjection.Variant Property
Gets a value that tells whether to use United States or International projection formulas.
Syntax
public StereographicVariant Variant { get; }Property Value
Default: UnitedStates
A StereographicVariant, either United States or International.
Remarks
Carmenta Engine does not support the spherical formulas for Stereographic, so the earth is always modelled as an ellipsoid, but there are two distinct variants that must not be confused. Although similar, their projected coordinates can differ by several meters: up to 8 meters within the Netherlands grid, for example.
(But in the polar aspect, with centralLatitude -90° or 90°, the two variants are the same, and they are also the same as PolarStereographicProjection.)
United States variant
This variant has been described by John P. Snyder of the U.S. Geological Survey (who credits P. D. Thomas of the U.S. Coast and Geodetic Survey). The earth ellipsoid is mapped in a conformal and continuous way to a Gaussian sphere, which is then mapped conformally to the plane by spherical formulas. This variant is hardly ever used for data storage. For example, there is no U.S. state plane coordinate system that is based on a stereographic projection. We think this variant would be forgotten now, if it had not been described in Snyder's excellent book.
In the PROJ library, this variant is known as +proj=stere.
International variant
This variant is described by EPSG in the OGP Guidance Note 7.2, and also in the manual for libproj4. As in the United States variant, the earth ellipsoid is first mapped conformally to a Gaussian sphere, which is then mapped conformally to the plane. But the step from ellipsoid to sphere is done in a different way, in order to minimize the distortion near the projection center. This requires more complicated formulas, but there are worse problems.
As the formulas are usually given, the ellipsoid surface is cut along the antipodal meridian, and the surface is then stretched in the west-east direction so that it overlaps itself slightly when it is wrapped around the sphere. The strip of overlap can be up to 1.2° wide on the Wgs84 ellipsoid; projected coordinates in the strip are ambiguous and therefore useless.
To avoid these problems, we have modified the international formulas so that they will shrink the ellipsoid surface on the backside of the earth, in order to compensate for the stretching on the front side. The advantage is that our projection is continuous (except at the antipode of the center point), so that it can be used for continental or hemisphere maps without worries. One drawback is that the projection is not exactly conformal on the backside of the Earth (farther than 90 degrees of longitude from the central meridian), but the deviation from conformality is quite small. Another drawback is that our formulas differ from the standard international formulas on the backside of the Earth. But the international stereographic is seldom used for data storage far from the center point, so we do not think this will be a problem in practice.
In the PROJ library, the international variant is known as +proj=sterea (where the "a" stands for alternative).
The international variant is also known as "Double Stereographic", which reflects the two-step process of projection (ellipsoid to sphere, and sphere to plane). Unfortunately, the U.S. variant is also a double projection. But in common usage, "Double Stereographic" usually means the International variant.
The Stereographic projection is also called "Oblique Stereographic", which says that aspect may be oblique (the projection center can be at any latitude). But both the U.S. and the International variant may be called Oblique.
What about Roussilhe?
There is a third variant known as Roussilhe stereographic, which Carmenta does not support. It was designed to be simpler to compute, in the days before electronic computers, and can be regarded as a very good approximation of the international variant. In some countries, like Poland, the national grids are officially based on Roussilhe stereographic. Then it is the other way around: the international variant becomes a very good approximation of Roussilhe. Near the projection center the differences are very small; they grow to about 5 centimeter at 350 km from the center.
In the PROJ library, Roussilhe stereographic is known as +proj=rouss.
Literature
John P. Snyder, Map Projections: A Working Manual (chapter 21). U.S. Geological Survey, Professional Paper No. 1395, Washington 1987.
IOGP Geomatics Guidance Note 7, part 2: Coordinate Conversions and Transformations including Formulas, section 1.3.7.
Gerald I. Evenden. libproj4: A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft), March 2005. http://dl.maptools.org/dl/libproj4
Platforms
Windows, Linux, Android