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.

Equations for direct mapping of the normal aspect are straightforward. Consider projecting point on Earth onto point 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 only, therefore .

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

Thus,

Making and using the trigonometric identities

for the range , the ordinates simplify to

Therefore,

Again, inverse mapping equations can be easily obtained.

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 just by reducing the cylinder's radius to . With a secant surface, the first proportion becomes

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

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

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

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

Consider a vertical 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 exacerbates the exaggeration of area
typical of high latitudes in cylindrical maps. In contrast, Lambert's
equal-area
cylindrical projection and its
variants preserve areas by
*compressing* instead of stretching latitudes,
but shapes suffer accordingly. This kind of compromise is a recurrent
theme of map projections.

www.progonos.com/furuti/MapProj/Normal/CartHow/HowBraunC/howBraunC.html — August 5, 2018

Copyright © 1996, 1997, 2012 Carlos A. Furuti