Definition of lenticular projections

General discussion of map projections.
PeteD
Posts: 251
Joined: Mon Mar 08, 2021 9:59 am

Re: Definition of lenticular projections

Post by PeteD »

daan wrote: Sat Feb 17, 2024 7:19 pm I just mean to observe that the number that has been described, let alone used commonly, is tiny in comparison
6.4 % of the pseudocylindrical projections listed on Tobias's website. The number is admittedly small but by no means negligible. While none of them are commonly used any more, the Times, Gall-Bomford and Hölzel projections were all once used in atlases, which is more than can be said of many projections.
daan wrote: Sat Feb 17, 2024 7:19 pm When I considered the lenticular category, I formalized the constraint rather than leaving it to natural selection. I don’t think that’s unfair
Maybe not unfair, but it's not really consistent.
daan wrote: Sat Feb 17, 2024 7:19 pm In a pseudocylindric projection, you don’t get the parallel curvature to alleviate distortion toward the poles, so you might be tempted to stretch the parallels apart to get a more conformal look.
I'm not convinced that the temptation to stretch the parallels apart is greater for pseudocylindricals. Compare Winkel II with the projection labelled as Winkel-Denner III on Tobias's website. It's just a variant of the Winkel Tripel where the averaging with the equirectangular projection is carried out only in the x direction. (I'm not trying to plug my own work -- it's the best example I can think of where two projections only really differ in that one of them has curved parallels.) The pseudocylindrical Winkel II has greater areal distortion towards the poles and wouldn't handle stretching the parallels apart very well at all. On the contrary, I'd be tempted to squash them together to get something more like the Winkel-Snyder.

On the other hand, while I don't think the lenticular Winkel-Denner III would be improved by stretching the parallels apart, it could get away with them being stretched apart better than the pseudocylindrical Winkel II could. This can be seen by comparing Winkel II with the projection labelled as Winkel-Denner V on Tobias's website. It's just a variant of Winkel-Denner III with increasing parallel spacing. Despite this increasing parallel spacing, Greenland is still smaller than in the pseudocylindrical Winkel II, which has constant parallel spacing, all thanks to the curved parallels.

In other words, since a projection with curved parallels has lower areal distortion towards the poles than a corresponding pseudocylindrical projection with the same parallel spacing, a slightly greater second derivative of y with respect to ϕ can be justified for the projection with curved parallels.
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