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95 points pheelicks | 2 comments | 13 May 25 10:02 UTC | HN request time: 0s | source
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xioxox ◴[13 May 25 15:36 UTC] No.43974102[source]▶
What's the advantage of this over HEALPix projection? https://en.wikipedia.org/wiki/HEALPix
replies(2): >>43975085 #>>43975193 #
pheelicks ◴[13 May 25 16:59 UTC] No.43975085[source]▶
The base platonic solid that Healpix is based on is the octahedron (https://en.wikipedia.org/wiki/Octahedron), which A5 uses the dodecahedron(https://en.wikipedia.org/wiki/Regular_dodecahedron).

The octahedron has a much higher angular defect (https://en.wikipedia.org/wiki/Angular_defect) than the dodecahedron, and thus when it is projected onto the sphere the cells are warped a lot. So while their areas may be the same, the shapes vary.

This article explains the geometric construction, and how it leads to the cells being a similar size and shape: https://a5geo.org/docs/technical/platonic-solids

Also from a data visualization point of view, the rectangular cells of Healpix (like S2) are arguably less pleasing to look at than hexagons/pentagons: https://h3geo.org/docs/comparisons/s2#visualization

replies(1): >>43977056 #
1. pama ◴[13 May 25 20:00 UTC] No.43977056[source]▶
Not sure I understand—healpix starts from the rhombic dodecahedron and then bisects the generalizations of the 12 squares each time. Where do octahedra come into play?
replies(1): >>43981304 #
2. pheelicks ◴[14 May 25 06:01 UTC] No.43981304[source]▶
My mistake, you are correct. The base solid is indeed the rhombic dodecahedron. I believe the point about the angular defect is still valid though.