OriginalsMarch 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.)
Figure 3 Lambert projection (1772), quality 0.3450
Figure 4 Hammer projection (1892), an elliptic version of Lambert's, quality 0.3758
Figure 5 Wagner VII projection (1941), a version of Hammer's with the pole as line, quality 0.2648
Better versionsThe criterion that people at the borders of the map are better served leads to different versions of existing projections
Figure 6 Neo-Lambert projection, quality 0.4141. This projection has the smallest circle containing all continents
Figure 7 Neo-Hammer projection, the center is undistorted, quality 0.4679
Figure 8 Neo-Wagner projection, the center has a prescribed distortion, like Wagner's original, quality 0.4675
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 projectionMathematically 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.
Figure 10 Cupola projection, quality 0.6803
Older family membersSeveral versions of the original three have been designed.
Figure 11 Gall-Peters projection (1885, 1967), quality 0.1691. The world is far from rectangular
Figure 12 Eckert-Greiffendorf projection (1935), quality 0.3160
Figure 13 Frančula-AK projection (1971), quality 0.2306
Figure 14 Frančula-A projection (1971), quality 0.3022
Figure 15 Aribert Peters' projection (1978), quality 0.2199
Figure 16 Canters version of Wagner's projection (2002), quality 0.1956
Figure 17 Boehm's version of Wagner's projection (2006), quality 0.3717
False solutionsWith 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.
Figure 18 Neo-Lambert false solution, quality 0.4104
Another WagnerWagner's projection has a prescribed distortion in the center. Without this prescription you'll get an ugly one.
Figure 19 Wagner-free projection, quality 0.5457
Ever more parametersThe 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!
Figure 20 Best projection with 1 parameter active, quality 0.5145
Figure 21 Best projection with 2 parameters active, quality 0.5412
Figure 22 Best projection with 3 parameters active, quality 0.6551
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 researchBecause 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.
Figure 24 Version of the cupola with horizontal asymmetry, same borders of the map, quality only marginally better, not significant: 0.6810
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
Figure 26 The same, but polar regions given their own map
Figure 27 The same, polar regions rotated
Figure 28 Latitudes above 70 north not taken into account, quality high 0.7235, but Greenleand very distorted
In practiceThanks 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'.
EquationsFor those who wants to have equations ('%' means: comment)
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)