Abetting a tattoo
Description of the tattoo’s projection
Having received inquiries about the tattoo’s projection, I will post an explanation here, since I have never published it formally.
If you consider the Mollweide and the Hammer projections, they both fill the same space and they are both equalarea projections. This implies that the transformation from the Mollweide to the Hammer is also an equalarea transformation. As long as a third projection fits within the space of the Mollweide, and as long as it is equalarea, then applying the MollweidetoHammer transformation to the third projection will yield a fourth, new equalarea projection.
This principle applies to any transformation of one equalarea projection to another, as applied to a third projection. Essentially it means you have deprojected the third projection back onto the sphere as if the plane coordinates of the third projection were actually the plane coordinates of the first projection. Then you project from the sphere to the plane via the second projection. This method of creating new equalarea projections does not appear in the map projections literature. I developed it in 1994 and presented it at that year’s meeting of the North American Cartographic Information Society in Ottawa.
My design criteria for this projection were that it be:
Without further ado, the generating formulæ are: With the usual convention that unsubscripted φ is latitude and unsubscripted λ is longitude. The subscripting on the intermediate variables reflects the transformation as I describe it above, where subscript e refers to Eckert IV and subscript p refers to the intermediate deprojection onto the sphere. A Newton’s iteration can be used to solve for θ.
Enjoy!
If you consider the Mollweide and the Hammer projections, they both fill the same space and they are both equalarea projections. This implies that the transformation from the Mollweide to the Hammer is also an equalarea transformation. As long as a third projection fits within the space of the Mollweide, and as long as it is equalarea, then applying the MollweidetoHammer transformation to the third projection will yield a fourth, new equalarea projection.
This principle applies to any transformation of one equalarea projection to another, as applied to a third projection. Essentially it means you have deprojected the third projection back onto the sphere as if the plane coordinates of the third projection were actually the plane coordinates of the first projection. Then you project from the sphere to the plane via the second projection. This method of creating new equalarea projections does not appear in the map projections literature. I developed it in 1994 and presented it at that year’s meeting of the North American Cartographic Information Society in Ottawa.
My design criteria for this projection were that it be:
 equalarea
 bilaterally symmetric
 in equatorial aspect, reduce the usual extreme distortions of the outer edges of land masses (Alaska, Siberia) by shoving into the Pacific the distortion they would normally accrue.
Without further ado, the generating formulæ are: With the usual convention that unsubscripted φ is latitude and unsubscripted λ is longitude. The subscripting on the intermediate variables reflects the transformation as I describe it above, where subscript e refers to Eckert IV and subscript p refers to the intermediate deprojection onto the sphere. A Newton’s iteration can be used to solve for θ.
Enjoy!
Re: Abetting a tattoo
I received an inquiry about how these projectiontoprojection mappings would look if:
Enjoy!
— daan
 HammerMollweide transformation were applied to the Mollweide
 MollweideHammer transformation were applied to the Hammer
Enjoy!
— daan
Re: Abetting a tattoo
Hi. Same Idhan as on the xkcd thread here! Thanks for those projections. Very interesting. The "double Mollweide" looks a little like a stretched out orthographic, while the "double Hammer" looks like a vertically compressed azimuthal equal area (which is, I suppose, what Hammer is from the start, but somehow the resemblance is stronger). I suppose that in principle there's no limit to how many iterations you could add to that process, although the maps might get pretty unrecognizable eventually. (Septuple Mollweide?) I don't really know much about map projections, though, other than reading Snyder's Flattening the Earth a few years ago.
Re: Abetting a tattoo
Those are evocative observations. The azimuthal equalarea’s outer boundary would represent the single point at the antipode of the center of the map, and so in that sense is topologically rather different, but I see where the resemblance comes in.Idhan wrote:The "double Mollweide" looks a little like a stretched out orthographic, while the "double Hammer" looks like a vertically compressed azimuthal equal area (which is, I suppose, what Hammer is from the start, but somehow the resemblance is stronger).
Uglier and uglier.Idhan wrote:I suppose that in principle there's no limit to how many iterations you could add to that process, although the maps might get pretty unrecognizable eventually. (Septuple Mollweide?)
That qualifies you as knowing quite a lot about map projections! That’s a fine, fine book. I heartily recommend it for anyone interested in the topic.Idhan wrote:I don't really know much about map projections, though, other than reading Snyder's Flattening the Earth a few years ago.
Re: Abetting a tattoo
I relaxed the topbottom symmetry constraint on my 1995 projection in order to substantially improve Australia and New Zealand.
This map uses the same 1995 projection for the northern hemisphere. For the southern, the map is the 1995 projection further modified as follows:
 Multiply x coordinates by 1.6 and divide y coordinates by the same.
 Deproject resulting coordinates to the sphere as if the mapped space is sinusoidal projection.
 Project from sphere to plane as McBrydeThomas II.
 Multiply x coordinates by ratio of sinusoidal equator to McBrydeThomas II; divide by 1.6.
 Divide y coordinates by ratio of sinusoidal equator to McBrydeThomas II; multiply by 1.6

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Re: Abetting a tattoo
Three years, and nobody ever answered?daan wrote: I’d like opinions on the æsthetics of such an asymmetric map.
So I give it a shot:
It's unusual, but I kinda like it because it’s unusual. I’m not sure if I'd recommend it if a publisher of school atlases (is that the correct english plural of "atlas"?) would ask me which projection should be used for the world map – but that doesn't matter because nobody ever asks me such things. I would hang up a wallpaper map using this projection, though, so visitors will ask me what kind of strange map this is and I can start a long discourse about map projections.
No, really, I like it. For one thing, because it's always good to be reminded that the world maps we are used to are only one way to project the earth – and for the other thing, because it simply looks cool.