Wed Jan 01, 2020 Two new projections, three variants, some adjustments
At the beginning of 2020, I’d like to publish a little update.
So, here are two new projections, three new variants – and a few
adjustments.
The BertinRivière projection
Based on a 1953 drawing by Jacques Bertin, which most likely was not created using a mathematical formula,
Philippe Rivière created this new projection in 2017.^{[1]}
… Well, I am calling this a new projection, because in my opinion it differs
too much from the original to be called Bertin projection.
So I’m using the name that is also used by the software G.Projector:
BertinRivière.
Mathematically, it was built using the Briesemeister projection^{[2]}, which itself was derived from the Hammer projection – so, hey, maybe I should call it the HammerBriesemeisterBertinRivière projection? 😉 Naaah…
However, it is an interesting projection that keeps the distortions of the land masses low. Enjoy:
And to emphasize the specific distribution of distortions, here’s an image with the Tissot indicatrix displayed on land masses only (the usual Tissot rendition is of course available at the list on the website).
And here’s the usual image of the Tissot indicatrix, along with the mathematical parent projection Briesemeister, and both the interrupted and uninterrupted version of SinuMollweide which is a bit similar.
Györffy E vs. BertinRivière
Györffy E vs. Briesemeister
Györffy E vs. SinuMollweide
Györffy E vs. SinuMollweide (uninterrupted)
Györffy E
In 2018, János Györffy introduced a series of five minimum error projections^{[3]}, labelled
A, B, D, E and F (C was discarded because there was no notable difference towards B).^{[4]} His goal was to reduce both angular and areal distortions on aphylactic (compromise) pointedpole projections with an outer shape
that will »remind the viewer of the Globe«.
A and B are pseudocylindricals while D, E and F belong the class of lenticular projections.
Regrettably, I can show but one projection out of this series, namely the Györffy E:
Let’s compare it to the Tissot indicatrix of some similar projections… ooops! There aren’t any!
Except for the Aitoff (which really doesn’t look similar), there are no lenticular pointedpole compromise projections! So, let’s
do the next best thing and compare it to some lenticular compromise projections with a pole line – there’s a bunch of them,
so I selected a few having areal distortions which are roughly in the ballpark of Györffy E’s distortion values. Very roughly.
You also can compare the Györffy E directly to the six other projections using the following links…
(Note: The links trigger the Expert Mode of the comparison. For these pairings, it’s
advisable to scroll down to the “scaled to same width” examples.)
Györffy E vs. Kramer VII
Györffy E vs. Winkel Tripel
Györffy E vs. Ginzburg V
Györffy E vs. Wagner IX.i
Györffy E vs. Wagner IX, Canters’ optimization
Györffy E vs. Wagner vii@65766040168
Three new variants
Also, I’ve added three variants of projections that already have been on this website for some time…
The equirectangular projection is already listed in two variants, one using the
equator as standard parallel (= plate carrée projection) and the other other one with
standard parallels at 35.6° N/S.
You’ll read below why I felt obliged to add another variant, this time with
standard parallels at 28° N/S.
The uninterrupted variant of Philbrick SinuMollweide. You’ve already seen it above – I have to admit, I can’t remeber why I neglected to add it the day I added the interrupted version.
Hey, a Wagner variant! I don’t think you’ve seen one before! 😉
This variant (already shown above, too) is somewhere in between the Wagner BCWA
and Canters optimization of Wagner IX. The Böhm notation^{[5]} is Wagner vii@65766040168 and for the lack
of creativity, I’m using this as the projection’s name.
There’s nothing special about this Wagner variant, I just… well, liked it.
To create this projection in Geocart,
choose the generalized Wagner with following parameters:
a = 2.16538, b = 1.4769, m = 0.975032, m2 = 0.75955, n = 0.422222
In d3geoprojection, use:
d3.geoWagner().poleline(65).parallels(76).inflation(40).ratio(168)
Lets have a glance at the distortion characteristics and then move on:
Adjustments
More than four years ago, I added the Patterson cylindrical projection – but the images I presented were wrong! 😱
There was a flaw in the map projection software I used for the Patterson images
(G.Projector)
so it was rendered with an widthtoheight ratio of roughly 0.61
instead of 0.57 as specified.
The flawed images were replaced by correct renditions.
Current versions of G.Projector create a correct Patterson as well.
And while I was at it, I replaced the topographic map in
the Patterson article
with a new one that I like much better.
🌐
The next adjustment can be handled in a single sentence: Images of the Dymaxion projection had broken graticule lines – thanks to a new version of Justin Kunimune’s map projection software, this is fixed.
🌐
Two years ago I began to use new images for the Tissot indicatrix, but only for projections that were added thereafter. Now, I replaced some of the older Tissot images with the new version. Namely all that were shown above (Winkel Tripel, Ginzburg V, …).
Footnotes

↑
Philippe Rivière: Bertin Projection (1953)
visionscarto.net/bertinprojection1953 
↑
Demonstrating the transformation of the Briesemeister to get the BertinRivière projection:
bl.ocks.org/Fil/5b9ee9636dfb6ffa53443c9006beb642 
↑
János Györffy: Minimum distortion pointedpolar projections for world maps by applying graticule transformation
doi.org/10.1080/23729333.2018.1455263  ↑ Györffy uses lowercase letters to label his projections, but I found uppercase letters to me more legible.

↑
Wagner’s Umbeziffern and the Böhm notation:
mapprojections.net/wagnerumbeziffern.php
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