CUPOLA PROJECTION AND ITS FAMILY




Originals

March 2021 an article of me has been published about a new small scale world map. Its projection is family of the Lambert projection, Hammer projection, Wagner VII projection and their descendants. These are uninterrupted equal-area projections, suited for all kinds of thematic world maps.
Aim was to find an equal-area projection where people at the borders of the continents are best served. Mathematically the smallest circumscribing irregular polygon around all continents has to be found. This brings the Pacific to the borders of the map and so Africa to the centre. It is accidentally that Europe is near the center of the map, it would have not been a problem for me at all when it had been otherwise. Antarctica is not taken into account.
The meridian of the left and right borders is chosen so that they do not cut land masses in two. This again leads to Europe and Africe laying in the centre, I can't help it.
Below are the figures of my article and extra ones. Each coloured dot represents 4 square degrees. Colours indicate distortion: dark red is undistorted, blue very much distorted. The quality of each projection is mentioned, namely the distortion of the most distorted points on the continents. Closer to 1 means less distorted. (The number represents the short semi-axis of Tissot's indicatrix.)

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Figure 3 Lambert projection (1772), quality 0.3450


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Figure 4 Hammer projection (1892), an elliptic version of Lambert's, quality 0.3758


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Figure 5 Wagner VII projection (1941), a version of Hammer's with the pole as line, quality 0.2648





Better versions

The criterion that people at the borders of the map are better served leads to different versions of existing projections

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Figure 6 Neo-Lambert projection, quality 0.4141. This projection has the smallest circle containing all continents


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Figure 7 Neo-Hammer projection, the center is undistorted, quality 0.4679


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Figure 8 Neo-Wagner projection, the center has a prescribed distortion, like Wagner's original, quality 0.4675


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Figure 9 Neo-Pécsi projection, quality 0.6429. Pécsi proposed a projection of the old world with a curved equator. This way to produce an asymmetric map can be used with my criterion for the whole world.





Best version: the Cupola projection

Mathematically five parameters are used for all projections of this family of equal-area projections. The above have zero to four parameters in use, with all five used the final map of this project is found.

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Figure 10 Cupola projection, quality 0.6803





Older family members

Several versions of the original three have been designed.

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Figure 11 Gall-Peters projection (1885, 1967), quality 0.1691. The world is far from rectangular


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Figure 12 Eckert-Greiffendorf projection (1935), quality 0.3160


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Figure 13 Frančula-AK projection (1971), quality 0.2306


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Figure 14 Frančula-A projection (1971), quality 0.3022


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Figure 15 Aribert Peters' projection (1978), quality 0.2199


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Figure 16 Canters version of Wagner's projection (2002), quality 0.1956


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Figure 17 Boehm's version of Wagner's projection (2006), quality 0.3717


False solutions

With 1 parameter active it is simple to find the best version ('best' in the sense of this project) with the help of a 2D graph. With 2 parameters active a 3D graph is helpfull. But even here false solutions can pop up (local maxima, not the sought for overall maximum). This happened for example by the search for the best version of Lambert of figure 6.

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Figure 18 Neo-Lambert false solution, quality 0.4104





Another Wagner

Wagner's projection has a prescribed distortion in the center. Without this prescription you'll get an ugly one.

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Figure 19 Wagner-free projection, quality 0.5457





Ever more parameters

The criterion I used (best borders of the continents) was useful for the Neo-projections above and for the final Cupola projection, but it also can lead to very ugly projections!

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Figure 20 Best projection with 1 parameter active, quality 0.5145


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Figure 21 Best projection with 2 parameters active, quality 0.5412


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Figure 22 Best projection with 3 parameters active, quality 0.6551


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Figure 23 Best projection with 4 parameters active, quality 0.6755. Here the asymmetry results not from Pécsi's way (figure 9) but from the curving of the equator I invented. This projection is already close to the Cupola projection, but the Cupola results from the two ways to achieve asymmetry together.


Extra research

Because there is much more land on the northern hemisphere asymmetry was allowed in my project, in the end leading to the Cupola projection with vertical asymmetry. Horizontal asymmetry also was possible in the computer programm.

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Figure 24 Version of the cupola with horizontal asymmetry, same borders of the map, quality only marginally better, not significant: 0.6810


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Figure 25 Version of the cupola with latitudes above 75 north not taken into account, because so few people are living there, quality 0.7052


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Figure 26 The same, but polar regions given their own map


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Figure 27 The same, polar regions rotated


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Figure 28 Latitudes above 70 north not taken into account, quality high 0.7235, but Greenleand very distorted


In practice

Thanks to several others the Cupola can already been found in libraries with many map projections.

mapshaper, click 'console', enter 'proj cupola densify' and enter 'graticule'.

d3-geo

Equations

For those who wants to have equations ('%' means: comment)
%central meridian:
c6=11.023 %[degrees], leads to borders -180+11.023 and 180+11.023

%when (lat,lon) or (phi, lambda) are [degrees], first make [radians]:
u=phi*pi/180; %phi is latitude in [degrees], u latitude in [radians]
v=lambda*pi/180 %lambda is longitude in [degrees], v longitude in [radians]

%x and y on the map; there is no scale factor, taken is earth radius=1
x=1.6188566*sqrt(2/(1+0.530815*sin(asin(0.7264*sin(u)+0.2587011))+0.8474875* cos(asin(0.7264*sin(u)+0.2587011))*cos(0.5253*v-0.1010612291)))* cos(asin(0.7264*sin(u)+0.2587011))*sin(0.5253*v-0.1010612291)*0.9701

y=1.6188566*sqrt(2/(1+0.530815*sin(asin(0.7264*sin(u)+0.2587011))+0.8474875* cos(asin(0.7264*sin(u)+0.2587011))*cos(0.5253*v-0.1010612291)))*(0.8474875* sin(asin(0.7264*sin(u)+0.2587011))-0.530815* cos(asin(0.7264*sin(u)+0.2587011))*cos(0.5253*v-0.1010612291))/0.9701



Weia Reinboud (weia (at) xmsnet (dot) nl)

Also see my page on all kind of things, in case it isn't open yet.