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The Cylindrical Equal-Area Projection

Although this projection is not particularly attractive in appearance, it has sometimes been seriously proposed for actual use, particularly in versions where the map is stretched vertically to a considerable extent.

Its main significance is as an intermediate mathematical step in the construction of some equal-area projections which we will examine later.

However, it can be noted that it doesn't look too bad within about 30 degrees of the Equator, and so it can be used in the transverse case for areas very narrow in longitudinal extent as well. This is useful not only for maps of Chile, or even Argentina, but also for things like gores used to make globes (although the Polyconic projection is the one most commonly used for that purpose) and also for maps showing the stars in the night sky that will be overhead at a certain hour of the evening at a certain time of the year. (For that purpose, though, I would be inclined to use the Gauss Conformal projection, the transverse case of the Mercator.)

This projection's equal-area properties are based on the fact, as shown in this diagram:

that the width of a degree of longitude is proportional to the cosine of the latitude, while the length of a degree of latitude remains constant.

In the Mercator projection, the latitude scale at any point was stretched by the same amount as the longitude scale had been stretched because the projection was cylindrical, to preserve shape.

In this projection, the latitude scale at any point is instead shrunk to make up for the fact that the longitude scale is stretched, so that the product of the two scales is constant, to preserve area.

Thus, in this case, the equation governing the position of the parallels is found by the following integral:

   |   cos(x), dx = sin(n)

which is a considerably simpler integral than the one that must be solved for the Mercator projection.

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