Comment by gct
19 days ago
I'd like to hear more about the synthetic part of these three spaces, because S2 works exactly as you say, embedding the 2-sphere in three (cartesian) dimensions. S2 points are always three dimensional.
19 days ago
I'd like to hear more about the synthetic part of these three spaces, because S2 works exactly as you say, embedding the 2-sphere in three (cartesian) dimensions. S2 points are always three dimensional.
S2 projects a 2-sphere onto a topologically homeomorphic 2-surface designed to play nicely with discrete representations. It is not a 3-space any more than a 2-sphere is. Most software uses three coordinates organized by a 2-surface DGGS.
DGGS that use 3-space embeddings are topologically 3-dimensional i.e. purely volumetric. They do not interpret the Earth as a 2-surface. In addition to polar coordinates, you must provide a volumetric model of the Earth to compute the DGGS cell. The shard distributions look very different between a 2-surface and a 3-surface. The latter has significantly better properties for large analytical data models but requires more sophisticated storage architectures.
The synthetic 3-space is optimized for two things. You want maximally efficient mapping function from the typical WGS84 geometry into it. Tidy math, basically. Since it is purely internal, the user will never see it, and it doesn't map to anything real, you have latitude to design it to satisfy software engineering objectives as long as it works. Second, the 3-space references are naturally less compact than 2-surface references at the same resolution even though you'll end up with roughly the same number of shards. A lot of effort goes to schemes to compress out the sparseness so that the storage requirements are similar to 2-surface DGGS e.g. how often do you need to represent geometry 1000 km below the Earth's surface?
These DGGS also have the low-key advantage that they natively represent and understand 3-space, not just surface geometry, if you move beyond making flat maps.
I'd be curious to hear more about how you do the 2 -> 3 embedding there. In S2 it uses cartesian three space, but points are constrained to be unit magnitude. This has advantage and disadvantages obviously.