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Juan de la Cosa’s Projection: A Fresh Analysis of the Earliest Preserved Map of the Americas
where subscript “0” indicates as usual the reference north-south meridian.
Using this formula for obliquity, we can rewrite the equations that define the Unnamed Projection, obtaining:
The UP can thus be visualized as the result of stretching north-south meridians and east-west meridians into a grid of vertical and horizontal straight lines equally spaced between them. This projection is therefore similar to the plate carrée or equirectangular projection except that horizontal lines represent constant obliquity instead of constant latitude.
Figures 11 and 12 show a sketch of the Atlantic basin drawn in the Unnamed Projection using two different reference meridians: 26ÂºW and 50ÂºW (0Âº = Greenwich).
It can be seen that, while north-south meridians are straight vertical lines, parallels of constant latitude appear as curves, all of them perpendicular to the reference meridian. As in the gnomonic projection, changing the reference meridian affects substantially the shape of the lands represented on the map, due to the fact that the vertical coordinate Y depends explicitly on the value of LON0. The horizontal dotted line represents the ecliptic, which naturally appears as a horizontal straight line in this projection.
A mathematical analysis shows that the UP is neither equivalent nor conformal, except for a single point: the origin of coordinates. Demonstrations can be found in annex 2. Nevertheless, the UP presents some interesting properties ( listed in Table 2) compared with those of other projections that were used or theoretically known in the early 16th century.
One of the advantages of the UP over cylindrical projections is that the pattern of deformation is equal along the east-west and north-south directions; in mathematical jargon, it presents a rotational symmetry of order 4. This property may be useful when drawing a world map because it does not give priority to north-south over east-west or vice versa.
In the UP, both vertical and horizontal lines are great circles. This is better than the plate carrée projection, where meridians are the only orthodromic lines drawn straight, but of course is inferior to the gnomonic projection, where all great circles are straight lines.
In theory the UP can represent a whole hemisphere, but in practice distortions become enormous for the regions near the North, South, West and East poles. Therefore, its use should be limited to an area reasonably close to the center of projection, say not more than 70 or 80Âº away.
Finally, the two main axes of coordinates of the UP are drawn in real magnitude. This is an advantage over azimuthal projections, in particular the gnomonic, which distorts scale badly along those axes.
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