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Deducing Braun's and other Stereographic Cylindrical Projections

Braun's Projection

Development of Braun's cylindrical map Schematic of Braun's cylindrical map
Development of Braun's stereographic cylindrical map, with the Eastern hemisphere on the left already unrolled. Light rays pierce the semitransparent globe drawing the right edge of the map. The light source on the Equator turns around the polar axis. Schematics of the globe and tangent projection surface. The map's height is twice the globe's diameter.

The stereographic cylindrical projection designed by C.Braun can be visualized geometrically, but it is slightly more complicated than the previous azimuthal orthographic example.  Instead of a flat projection plane directly yielding the projected map, here the projection surface is a cylindrical sheet tightly rolled against the Equator. Every meridian is drawn on this tube by light rays emanating from an equatorial point on the meridian directly opposite. The tangent tube is then cut along an arbitrary meridian and unrolled.

There is a similar procedure for creating some cylindrical projections, although not all of them employ such a simple model. In fact, well-known projections like Mercator's and the equidistant cylindrical are defined by arbitrary constraints, not a perspective process.

Braun's stereographic cylindrical map
Map in Braun's stereographic cylindrical projection

Equations for direct mapping of the normal aspect are straightforward. Consider projecting point P on Earth onto point P' on the map. First, like in all cylindrical designs, meridians in this aspect are projected as straight vertical lines with spacing directly proportional to longitude lambda only, therefore x=Rlambda.

The parallel determined by phi is projected as a straight horizontal line, whose y-coordinate can be derived from the diagram by a simple proportion:

h/(w+R)=y/2R

Thus,

h=Rsin phi, w=Rcos phi, y=2Rsin phi/(1+cos phi)

Making theta=phi/2 and using the trigonometric identities

sin(2theta)=2sin theta cos theta; cos(2theta)=1-2sin^2 theta
for the range -pi2<=phi<=pi/2, the ordinates simplify to
y=2R2sin theta cos theta/(1+1-2sin^2 theta)=4Rsin theta cos theta/2(1-sin^2 theta)=2Rsin theta cos theta/cos^2 theta
Therefore,
x=lambda R, y = 2R tan(phi/2)

Again, inverse mapping equations can be easily obtained.

Gall's Stereographic and Others

On Braun's projection, the cylindrical surface is tangent and the Equator is the standard parallel. The derivation above can be immediately generalized to a different standard parallel lambda0 just by reducing the cylinder's radius to R cos phi0. With a secant surface, the first proportion becomes

R sin phi/R(1+cos phi)=y/R(1+cos phi0)

Thus the equations for stereographic cylindrical projections like Gall's (phi0 = 45°) and the BSAM cylindrical (phi0 = 30°) are

x=R cos phi0 lambda, y = R (1+cos phi0)tan(phi/2)

Further Generalizations: the pseudo-Mercator and the central cylindrical

Finally, it's possible shifting the light source from a point on the Equator, i.e., at a distance R from the polar axis, to a distance kR. The equations, without the trigonometric substitution above, become

x=R cos phi0 lambda, y = R (k+cos phi0)sin phi / (k + cos phi)

Two particular cases are Braun's pseudo-Mercator projection with k=0.4 and the central cylindrical projection, with k=0 (both use phi0 = 0 = 0°).

Trading shape for area: the eternal cartographic compromise
Braun's stereographic cylindrical map
Oblique azimuthal ortographic map superimposed onto partial Braun's stereographic cylindrical equatorial map, drawn at identical scaling factors. Selected 10°-wide graticule cells highlight how most cylindrical maps exaggerate height (therefore area) towards the poles in order to better preserve shape. Braun's exaggeration is noticeable, but pales in comparison with Mercator's

Across all cylindrical projections, the only difference is the vertical scale, which in the equatorial aspect translates to the spacing between parallels. Most of them, like Braun's and Gall's stereographic, are designed to increase the spacing towards the poles. Why?

Consider a series of graticule "cells" of uniform "width" and "height" (say 10°); on Earth, they are approximately square at the Equator but with growing latitudes they progressively narrow down, becoming very thin spherical triangles at the poles. By definition, cylindrical projections cannot narrow their cells, only stretch them vertically as an approximation. In particular, in Mercator's the stretching is calculated in order to exactly preserve local angles; this requires the poles to be projected into infinity.

Of course, vertical stretching can only exacerbate the exaggeration of area typical of high latitudes in cylindrical maps. In contrast, Lambert's equal-area cylindrical projection and its variants preserve area by compressing instead of stretching latitudes, but shapes suffer accordingly.



HomeSite MapAzimuthal OrthographicHow Projections are CreatedKavrayskiy VII  www.progonos.com/furuti    October 30, 2014
Copyright © 1996, 1997, 2012 Carlos A. Furuti