Abetting a tattoo

General discussion of map projections.

Abetting a tattoo

Postby daan » Mon Nov 14, 2011 10:20 pm

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Description of the tattoo’s projection

Postby daan » Mon Nov 21, 2011 11:57 am

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 equal-area projections. This implies that the transformation from the Mollweide to the Hammer is also an equal-area transformation. As long as a third projection fits within the space of the Mollweide, and as long as it is equal-area, then applying the Mollweide-to-Hammer transformation to the third projection will yield a fourth, new equal-area projection.

This principle applies to any transformation of one equal-area 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 equal-area 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:
  • equal-area
  • 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.

To achieve this, I chose the Eckert IV projection, which is also equal-area, as the projection to apply the Mollweide-Hammer transformation to. I scale the Eckert IV to fit within the confines of the Mollweide, and then transform the results using Mollweide-to-Hammer. The amount of scaling can be adjusted to modify the appearance of the projection. The default scaling as implemented in Geocart is the parameterization I chose for “most pleasing” results within the constraints of the design criteria.

Without further ado, the generating formulæ are:
Strebe_1995.png
Generating formulæ for Strebe 1995.
Strebe_1995.png (39.23 KiB) Viewed 11219 times


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!
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Re: Abetting a tattoo

Postby daan » Fri Nov 25, 2011 3:13 am

I received an inquiry about how these projection-to-projection mappings would look if:
  • Hammer-Mollweide transformation were applied to the Mollweide
  • Mollweide-Hammer transformation were applied to the Hammer

These are they.

double_Mollweide.jpg
Hammer-Mollweide transformation applied to the Mollweide
double_Mollweide.jpg (169.96 KiB) Viewed 11217 times


double_Hammer.jpg
Mollweide-Hammer transformation applied to the Hammer
double_Hammer.jpg (182.5 KiB) Viewed 11217 times


Enjoy!
— daan
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Re: Abetting a tattoo

Postby Idhan » Fri Nov 25, 2011 11:28 pm

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.
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Re: Abetting a tattoo

Postby daan » Sat Nov 26, 2011 3:27 am

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).


Those are evocative observations. The azimuthal equal-area’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: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?)


Uglier and uglier. ;)

Idhan wrote:I don't really know much about map projections, though, other than reading Snyder's Flattening the Earth a few years ago.


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.
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Re: Abetting a tattoo

Postby daan » Mon Nov 28, 2011 7:58 pm

I relaxed the top-bottom symmetry constraint on my 1995 projection in order to substantially improve Australia and New Zealand.
2011.jpg
Modified southern hemisphere to improve Australia/New Zealand.
2011.jpg (328.49 KiB) Viewed 11163 times

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 McBryde-Thomas II.
  • Multiply x coordinates by ratio of sinusoidal equator to McBryde-Thomas II; divide by 1.6.
  • Divide y coordinates by ratio of sinusoidal equator to McBryde-Thomas II; multiply by 1.6
The result is equal-area, of course. I’d like opinions on the æsthetics of such an asymmetric map.
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Re: Abetting a tattoo

Postby Atarimaster » Sat Dec 13, 2014 9:35 am

daan wrote:I’d like opinions on the æsthetics of such an asymmetric map.


Three years, and nobody ever answered?
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. :twisted:

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.
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