Previous Section	Projection Formula Table of Contents	Next Section

3.4.4.6 The Stereographic and Polar Stereographic

The Stereographic projection may be imagined to be a projection of the earth's surface onto a plane in contact with the earth at a single tangent point from the opposite end of the diameter through that tangent point.

This projection is best known in its polar form and is frequently used for mapping polar areas where it complements the Universal Transverse Mercator used for lower latitudes. Its spherical form has also been widely used by the US Geological Survey for planetary mapping and the mapping at small scale of continental hydrocarbon provinces. In its transverse or oblique ellipsoidal forms it is useful for mapping limited areas centered on the point where the plane of the projection is regarded as tangential to the ellipsoid., e.g. the Netherlands. The tangent point is the origin of the projected coordinate system and the meridian through it is regarded as the central meridian. In order to reduce the scale error at the extremities of the projection area it is usual to introduce a scale factor of less than unity at the origin such that a unit scale factor applies on a near circle centered at the origin and some distance from it.

The coordinate transformation from geographical to projected coordinates is executed via the distance and azimuth of the point from the center point or origin. For a sphere the formulas are relatively simple. For the ellipsoid the same formulas are used but with auxiliary latitudes, known as conformal latitudes, substituted for the geodetic latitudes of the spherical formulas for the origin and the point .

Oblique and Equitorial Cases

Given the geodetic origin of the projection at the tangent point (j₀, l₀), the parameters defining the conformal sphere are:

where:

The conformal latitude and longitude (c₀, L₀) of the origin are then computed from:

where S₁ and S₂ are as above, and

Then, for any point with geodetic coordinates (j, l) the equivalent conformal latitude and longitude (c, L) are compute from

, and

where:

Then, we find

which results in

The reverse formulas to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values.

The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates ( E, N ):

where

Geodetic longitude:

Isometric latitude:

First approximation:

where B = base of natural logarithms

y_i = isometric latitude at j_i

where

Then iterate:

until convergence.

If a projection is the equitorial case, j₀ and c₀ will be zero degrees and the formulas are simplified as a result, but the above formulas remain valid.

For the polar version, j₀ and c₀ will be 90 degrees and the formulas become indeterminate. See below for the formulas for the polar case.

For Stereographic projections centered on points in the southern hemisphere, including the south Polar Stereographic, the signs of E, N, l₀, l, must be reversed to be used in the equations. j will be negative anyway as a southerly latitude.

Example:

For Projected Coordinate System RD / Netherlands New

Parameters:

Forward calculation for:

first gives for conformal sphere constants:

where

For the point (j, l), c = 0.924394997, L = 0.104724841 rad.

hence, b = 1.999870665, and

Easting (E) = 196105.283 m

Northing (N) = 557057.739 m

An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed. The conformal longitude is then always equivalent to the geodetic longitude. This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered by EPSG to be a different projection method.

Reverse calculation for the same easting and northing first gives:

then L = 0.10472467, whence l = 0.104719584 rad = 6° E.

Also, c = 0.924394767, and y = 1.089495123

Iterate, beginning with these values. Then

j1 =	0.921804948				y1 =	1.084170164
j2 =	0.925031162				y2 =	1.089506925
j3 =	0.925024504				y3 =	1.089495505
j4 =	0.925024504

leads to (after convergence)

j = 0.925024504

Then

Latitude j = 53°00' 00.00" N

Longitude l = 6°00' 00.00" E

Polar Case

For the forward transformation from latitude and longitude,

where

For the reverse transformation,

where

Previous Section	Projection Formula Table of Contents	Next Section