# Projection planes that conform to the surface

The ISEA projection and other global ones (see DGGS standard) showed that the technology exists: it is possible to use a "coarse cover of polygons" (typically triangles, squares or hexagons), to cover the Earth's crust, and use each polygon as a plane of projection. ISEA and other (ex. S2 Geometry) offers a continuous "projection cover", with no fitting problems — no border problem like in the old UTM systems.

So we can think of a "partial cover", without topological (spherical) compromise... The main question is "Is there a rigorous methodology (or software tools) for build it?"

Rationale: useful as "cover of projection planes" in a continental country such as Russia or Brazil. Each plane with the local mean altitude, and to build a hierarchical grid of equal-area cells.

Illustrating with an intuitive home example:

ISEA and other projections are global, and are analogous to the "fit to plane" tiles (a set of planes that fit to sphere); I need a non-global and "fit to local surface" projection technology — each plane with a local median altitude.

Bellow a concrete example, covering Brazil with 4 squares of ~3000 km of side: each with angular freedom to a local best fit to Brazilian's surface.
The labeled squares form 3 pairs connected by "hinges": 1-2; 2-3; 2-4.

PS: the supposed gain in accuracy with this technique could be compared to the Albers equal-area projection for full territory... perhaps no big gain, but I need to check.

Bellow other example, covering Brazil with 7 hexagons: the main ones (labeled 1, 3, 4 and 6) each with angular freedom to a local best fit to Brazilian's surface, and the secondary ones (2, 5 and 7) where only a litle red portion are members of the country, with no border conflict with other hexagons. All connected hexagons has angular freedom: 1-3; 1-7; 3-2; 3-4; 3-5; 3-6.

## Notes

• (edit after @mkennedy answer) in the LDP (low-distortion projections) jargon, we can say "map projection zones have zone overlap"... And LDP initiatives are not trying to solve the "span across multiple projection zones" problem.

• This question is not about arbitrary mosaic of projection zones, but about the use of a limited number of projection zones, that covers the country and have no border problem: a polygon can span across multiple projection zones, since they are into the country.

• Here a link to Wikipedia describing the classic overlapping UTM grid zones problem (and here a question about it). This question is about a technology that avoids the problem.

• See part of the solution at gis.stackexchange.com/q/418669/7505 . A solution is to use a kind of equivalent-area global projection (like ISEA), them correct by altitude, with de mosaic of planes of the question. Commented Feb 3, 2022 at 11:50