Tetrahedral projections

AuthaGraph


Top: The official AuthaGraph map
Above: The projection's clone applied to a silhouette map of the Earth

Pros:

  • Minimum-error projection: balances out areal and angular distortions to an acceptable degree.
  • Good placement of landmasses to avoid severe distortions.
  • Won the Japanese Good Design Grand Award in 2016.
  • Probably the most well-known tetrahedral projection out of all after it took the media by storm.

Cons:

  • Its equations is proprietary and has never been published anywhere, requiring people to resort to alternatives such as its reverse-engineered clone to make similar maps.
    • Because of this, it’s impossible to use the original projection in any GIS software.
    • Not only that, every single image of this projection on the internet uses THE SAME SHITTY LOW-RESOLUTION POLITICAL MAP, WHICH IS THE ONLY MAP THEY WERE WILLING TO PUBLISH ON THEIR WEBSITE (see above). UPDATE: I’ve managed to find a high resolution image of the map. See here.
  • Landmasses such as Australia or Brazil look weird with this projection despite the its minimum-error properties.
  • Compromise projection means that it’s not really equal-area as it name claims it to be.

Lee tetrahedral

Pros:

  • One of the best conformal projections: preserves angles locally while maintaining area proportions relatively well unlike most conformal projections. A good configuration of the map can yield results where most landmasses avoid touching the singularities, where its conformality is lost and distortions are the worst.
    • The AuthaGraph-like configuration is a good example of this.
  • Unlike most conformal projections, it’s possible to show the entire world with this projection, and it tessellates too.
  • Generally looks more appealing than compromise or equal-area tetrahedral projections due to its angle-preserving nature, and landmasses look more “normal”.

Cons:

  • Higher area distortion than compromise projections, although this is to be expected.
    • This causes Western Australia to appear noticeably inflated.
  • As with most conformal projections, singularities do exist on the edges of the map.

van Leeuwen

Pros:

  • Equal-area: absolutely no areal distortions unlike AuthaGraph.
  • Using the AuthaGraph-like configuration also yield pretty good results to minimize its angular distortions.

Cons:

  • Severe angular deformations are noticeable with this projection, though this is to be expected due to its authalic properties.
  • Landmasses look even weirder than AuthaGraph because of this.

Kunimune’s “EquaHedral”

Pros:

  • Interrupted equal-area: Minimizes distortions that is caused by its area equivalence properties, resulting in a better looking map than van Leeuwen.
  • Customizable: Sinus length can be adjusted.

Cons:

  • Interruptions make it difficult to measure distances in certain cases, and can slice up landmasses.
  • Its best configuration with sinus length of 42 isn’t compatible with the AuthaGraph-like positioning of landmasses, since it slices up Brazil, Russia, and Indonesia.

Kunimune’s “TetraGraph”