Blended equal-area projections

General discussion of map projections.
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daan
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Blended equal-area projections

Post by daan »

As I noted in another posting here, I have developed a method for blending equal-area projections, yielding equal-area results.

Here is a sample sequence, and here is higher resolution.
Lambert to Albers, parameterized [0..1] in 0.1 increments.
Lambert to Albers, parameterized [0..1] in 0.1 increments.
LambertToAlbersThumbnail.jpg (288.78 KiB) Viewed 2123 times
Notice that the topologies of the first and final projections are distinct. The blending still works fine. Determing the outer boundary isn’t trivial when the topologies differ.

The first projection in Lambert azimuthal equal-area centered at 37°4′24″. The last is Albers with standard parallels of 29°30′ and 45°30′.

Enjoy!
— daan
Atarimaster
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Joined: Fri Nov 07, 2014 2:43 am

Re: Blended equal-area projections

Post by Atarimaster »

This really looks interesting!
Thanks for sharing, I’m looking forward to try it in Geocart.

Kind regards,
Tobias
daan
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Re: Blended equal-area projections

Post by daan »

daan
Site Admin
Posts: 977
Joined: Sat Mar 28, 2009 11:17 pm

Re: Blended equal-area projections

Post by daan »

Atarimaster wrote:This really looks interesting!
Thanks for sharing, I’m looking forward to try it in Geocart.

Kind regards,
Tobias
Actually, you already can. See the diagram of the method here:
https://www.mapthematics.com/Downloads/Progression.png

You have projections A and B and you want a projection somewhere between them, with k being the “distance” between A and B, and 0 ≤ k ≤ 1.
  1. Project to A. Use high resolution inversely proportional to k. If k < ½, consider going from B to A instead.
  2. Scale projected A by k.
  3. Export your kA projection as a TIFF image, preserving the high resolution.
  4. Place the kA image on a new document.
  5. Make a new map A at the original scale. Center A on kA.
  6. Export the map as a reprojection database.
  7. Make a new map in projection B. Add the reprojection database, and turn off the graticule and outer boundaries.
  8. Scale the new map by 1/k.
In Step 5, I say to “center” the map, but it’s more nuanced than that. If you don’t want any more steps involved, then you need to find the point in A that has no distortion and overlay it onto the image at the same place. However, the process can still work even if the point you want to be the “center” has distortion. It just requires more steps involving scaling.

It’s a manually intensive process, but you can experiment to heart’s content. Let me know if you need more information on how to do this.

Best,
— daan
daan
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Joined: Sat Mar 28, 2009 11:17 pm

Re: Blended equal-area projections

Post by daan »

The peer-reviewed paper is now published online in Cartography and Geographic Information Science. You can acquire the Author's Manuscript here. The Version of Record is here. The print edition will come out in a few months.

Cheers!
— daan
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