OGC Topic 21 Conformance

The Open Geospatial Consortium (OGC) describes discrete global grid systems (DGGS) in its Abstract Specification Topic 21 (OGC 20-040r3, consistent with ISO 19170-1). Topic 21 defines a DGGS as a hierarchy of grids that each cover the globe once, zonal identifiers that name the cells (zones), and functions to assign data to zones, query them and exchange results. Its 29 requirements fall into three conformance classes: common spatio-temporal classes (Requirements 1-5), the DGGS core (6-19) and the Equal-Area Earth Reference System, EAERS (20-29). This page sets hexify against each of them, gives the measurements a few of them ask for, and shows the functions that answer Topic 21’s zone queries.

Installation

# Install from CRAN
install.packages("hexify")

# Or install development version from GitHub
# install.packages("pak")
pak::pak("gcol33/hexify")

Earth model and area error

Topic 21 lets a DGGS sit on a sphere or an ellipsoid; the misfit of a sphere to the Earth goes into an area error budget, which Requirement 28 caps at 1%. hexify builds an Earth grid on a sphere whose area is that of the WGS84 ellipsoid, and reads geodetic latitude as latitude on that sphere. On the sphere, Snyder’s projection makes every hexagon of a resolution the same area (pentagons 5/6 of it). On the ellipsoid, a small region at geodetic latitude $\varphi$ then has $MN/R_q^2$ times its spherical area, with $M$ and $N$ the meridional and prime-vertical radii of curvature and $R_q$ the authalic radius:

a <- 6378137; f <- 1 / 298.257223563; e2 <- f * (2 - f); e <- sqrt(e2)
q <- function(phi) (1 - e2) * (sin(phi) / (1 - e2 * sin(phi)^2) -
                                 log((1 - e * sin(phi)) / (1 + e * sin(phi))) / (2 * e))
rq2 <- a^2 * q(pi / 2) / 2
area_scale <- function(lat_deg) {
  s2 <- sin(lat_deg * pi / 180)^2
  a^2 * (1 - e2) / (1 - e2 * s2)^2 / rq2
}
round(100 * (area_scale(c(0, 30, 45, 60, 90)) - 1), 3)
#> [1] -0.447 -0.113  0.223  0.560  0.899

A cell’s area on WGS84 therefore lies between 0.45% below and 0.90% above the sphere’s, by latitude alone: inside the 1% budget. The cells of every resolution still tile the ellipsoid exactly, since the sphere and the ellipsoid have the same area. paper/bench/bench_ogc_conformance.R reads the same factor at the area centroid of every cell of ISEA3H, ISEA4H and ISEA7H grids, with the same extremes. The DGGRS definitions OGC registers for ISEA3H and ISEA7H instead convert geodetic latitude to authalic latitude, which makes cells equal-area on the ellipsoid itself (issue #104).

Fuller’s projection (projection = "fuller") is not equal-area, so a Fuller grid falls outside the EAERS class (Requirements 22 and 29), as does H3.

Requirements

“Has” means hexify meets the requirement, “partial” that it meets part of it, “lacks” that it does not. Requirements 2, 3, 5 and 20 ask that a system’s data model follow Topic 21’s UML diagrams; hexify is an R library whose objects were not built from them, and they are not assessed here.

Req	Asks for	hexify	How
1	Temporal geometry and topology	lacks	Zones are spatial only
2	Zonal geometry and topology classes	not assessed	UML data model
3	Spatial location classes	not assessed	UML data model
4	Temporal reference system by period identifiers	lacks	Zones are spatial only
5	Reference system by zonal identifiers	not assessed	UML data model
6	Core reference system data model	has	`hex_grid()` holds solid, aperture, resolution, orientation, projection, radius and CRS; `dggrs_definition()` adds refinement strategy and grid constraints
7	A CRS, with coordinate epoch	partial	The grid’s `crs`, WGS84 on Earth; no coordinate epoch
8	A global domain and its dimension	has	The whole sphere of `radius_km`, two-dimensional
9	Level 0 covers the domain	has	The twelve vertex cells of the icosahedron; `n_cells()` counts every level
10	Every location in exactly one level-0 zone	has	Point assignment is a function; every cell corner is shared by exactly three cells
11	Simple zone geometry	has	Hexagons and twelve pentagons (squares on the octahedron)
12	A position inside each zone	has	`cell_to_lonlat()`, the cell centre
13	Globally unique zonal identifiers	has	`cell_to_index()` strings carry their resolution (see below)
14	A grid is all zones of one level	has	One `hex_grid()` is one level
15	Grids ordered by refinement	has	Resolutions 0-30 (ISEA), 0-15 (H3)
16	Quantization functions	partial	`hexify()`, `lonlat_to_cell()`, `hex_summarize()`, `hex_zonal()`, `hex_extract()`, `hex_compact()`; no tile roles
17	Zone queries	partial	`get_parent()` (one parent or all overlapping), `get_children()`, `get_siblings()`, `get_neighbors()` (disk or ring), `hex_distance()`; no DE-9IM predicates on zone sets
18	Read and run external queries	lacks	Queries are R function calls
19	Translate and broadcast results	partial	`cell_to_sf()` (any GDAL format), `as_dggrid()`, `from_dggrid()`, `import_h3()`, `dggrs_definition()`; no transport
20	EAERS data model	not assessed	UML data model
21	Domain is the Earth model’s surface; equal-sized cells	has	Sphere of WGS84 area; error above
22	Initial equal-area tessellation from a base polyhedron	has (Snyder)	Fuller is not equal-area
23	Progressively smaller cells	has	Apertures 3, 4, 7 and any sequence of them
24	Limit on refinement tied to the area budget	partial	Limit set by 64-bit cell IDs (resolution 30, 29, 21), not by the budget
25	Complete and unique at every level	has	As 9 and 10, at every resolution
26	EAERS cells are simple polygons	has	As 11
27	Position is the area centroid	partial	The cell centre, within a small offset of the centroid (below)
28	Area error budget of 1% or less	has	-0.45% to +0.90% on WGS84 (above)
29	Equal area within the budget	has (Snyder)	Exact on the sphere; Fuller and H3 are not equal-area

Zonal identifiers

A cell ID numbers the cells of one resolution from 1, as DGGRID’s SEQNUM does, so it names a cell only together with its grid. The strings cell_to_index() writes carry the resolution in their length, so they name one cell among all resolutions and serve as the zonal identifier of Requirement 13: Z7 (IGEO7) for aperture 7, Z3 for aperture 3, Z-order for aperture 4, and a two-digit-per-level string for mixed sequences. Each is hierarchy-based, one of the indexing methods Topic 21 lists (8.2.4.3).

g7 <- hex_grid(resolution = 5, aperture = 7)
cell <- lonlat_to_cell(16.37, 48.21, g7)
cell
#> integer64
#> [1] 35965
cell_to_index(cell, g7)
#> [1] "0003246"
cell_to_index(get_parent(cell, g7), hex_grid(resolution = 4, aperture = 7))
#> [1] "000324"

Zone queries

Topic 21’s parent() returns every coarser zone a zone overlaps, or with inheritID = true only the one its identifier descends from. hexify’s default parent is the coarser cell holding the cell’s centre, the inheritID = true answer; overlapping = TRUE returns all of them. The hexagons of successive resolutions do not nest: an aperture-3 cell on a parent’s corner lies a third in each of three parents, an aperture-4 cell on a parent’s edge half in each of two, and an aperture-7 cell off the parent’s centre 11/12 in its parent and 1/12 in one neighbour.

g3 <- hex_grid(resolution = 6, aperture = 3)
near <- get_neighbors(lonlat_to_cell(16.37, 48.21, g3), g3, include_self = TRUE)[[1]]
get_parent(near, g3)
#> integer64
#> [1] 532 532 532 523 541 747 747
ov <- get_parent(near, g3, overlapping = TRUE)
lengths(ov)
#> [1] 1 3 3 3 3 3 3
ov[lengths(ov) == 3][1]
#> [[1]]
#> integer64
#> [1] 532 514 523

Siblings are the other children of a cell’s parent, and a hollow ring the cells exactly k steps away:

get_siblings(cell, g7)
#> [[1]]
#> integer64
#> [1] 35916 36015 36064 36112 36162 36211
get_neighbors(cell, g7, k = 2, ring = TRUE)
#> [[1]]
#> integer64
#>  [1] 35671 35720 35769 35768 35915 35868 36015 36063 36161 36162 36211 36259

Topic 21 gives the refinement ratios of a DGGS as a list that recurs when it is shorter than the hierarchy. hex_grid() reads a short aperture list the same way:

hex_grid(resolution = 5, aperture = c(4, 3))@aperture
#> [1] "4,3,4,3,4"

Centroid

Requirement 27 asks that a zone’s position be its centroid, “computed as the geodesic centre of surface area”. cell_to_lonlat() gives the inverse projection of the centre of the planar hexagon, which is the area centroid where a cell is symmetric on the sphere and lies close to it elsewhere. paper/bench/bench_ogc_conformance.R measures the offset: the area centroid of a region of the sphere points along half the integral of $\mathbf{x} \times d\mathbf{x}$ around its boundary.

Grid	Resolution	Centre spacing (km)	Median offset (km)	Largest offset (km)	Largest / spacing
ISEA3H	3	1480	4.41	72.4	0.049
ISEA3H	6	286	0.29	16.4	0.057
ISEA3H	9	55.0	0.0088	3.30	0.060
ISEA3H	12	10.6	0.00032	0.627	0.059
ISEA4H	4	482	0.93	26.8	0.056
ISEA4H	8	30.2	0.0026	1.79	0.059
ISEA7H	3	417	0.51	22.3	0.054
ISEA7H	5	59.5	0.010	3.54	0.059
ISEA7H	7	8.51	0.00021	0.510	0.060

Grids up to a few thousand cells are measured whole; finer ones on the cells holding 20,000 uniform points. The median offset falls by about two orders of magnitude every time the cells shrink tenfold, as expected where the projection varies smoothly across a cell. The largest offset stays near 6% of the distance between neighbouring centres at every resolution. For data that need the area centroid as the zone’s position, the centre is therefore an approximation, within the stated fraction of a cell.

Binning and the hierarchy

Because the hexagons do not nest, a point’s cell at a fine resolution and its parent do not always agree with the point’s cell at the parent’s resolution. Griffin (2026) measures this on 10,000 points. On the lattice the rate one level up follows from the shares above: aperture 3, $2/3 \cdot 2/3 = 4/9$ ; aperture 4, $3/4 \cdot 1/2 = 3/8$ ; aperture 7, $6/7 \cdot 1/12 = 1/14$ . Snyder’s projection is equal-area, so the sphere gives the same rates up to the cells beside the twelve vertices. paper/bench/bench_hierarchy_commutation.R measures it on 10,000 uniform points:

Grid	Lattice, one level	1 level	2	3	4	5	6
ISEA3H, resolution 12	4/9 = 0.444	0.452	0.473	0.448	0.604	0.455	0.651
ISEA4H, resolution 10	3/8 = 0.375	0.371	0.528	0.599	0.632	0.654	0.651
ISEA7H, resolution 8	1/14 = 0.071	0.070	0.061	0.066	0.067	0.065	0.066
FULLER7H, resolution 8	1/14 = 0.071	0.077	0.064	0.068	0.066	0.067	0.065
H3, resolution 8	1/14 = 0.071	0.074	0.066	0.068	0.066	0.066	0.068

The standard error of each rate is about 0.005 (0.0025 for aperture 7). One level up, each equal-area grid gives its lattice rate within two standard errors. Further up, aperture 7 stays near 6.5%, as Griffin (2026) reports for H3; apertures 3 and 4 rise to about two thirds. One level up, the parent from get_parent() is the cell a point falls in at the coarser resolution for 93% of points at aperture 7, 55% at aperture 3 and 63% at aperture 4; an aggregation that must follow the coarser grid exactly bins the points at that resolution directly.

Area and shape against other grids

Kmoch et al. (2022) compared the normalized area (a cell’s area over the mean) and the isoperimetric quotient $4\pi A/p^2$ of ten open-source DGGS, reading each cell in a Lambert azimuthal equal-area plane on WGS84 centred on the cell. cell_metrics() returns both, measured on the sphere, as normalized_area and ipq. paper/bench/bench_kmoch_distortion.R runs their pipeline on hexify’s cells and sets the result beside their tables:

Grid	Resolution	Cells	SD of normalized area, this pipeline	Kmoch et al.	Mean ipq, this pipeline	Kmoch et al.	SD of normalized area on the sphere
ISEA3H	6	7,292	0.0045	0.0040	0.8964	0.8964	0.0068
FULLER3H	6	7,292	0.0291	0.0287	0.9015	0.9015	0.0303
ISEA4H	7	163,842	0.0040		0.8964		0.0014
ISEA43H	8	207,362	0.0040		0.8964		0.0013
ISEA7H	4	24,012	0.0040	0.0040	0.8964	0.8964	0.0037
ISEA7H	5	168,072	0.0048	0.0043	0.8968	0.8969	0.0014
FULLER7H	5	168,072	0.0295	0.0290	0.9010	0.9010	0.0292
H3	4	288,122	0.1334	0.1306	0.9047	0.9047	0.1334

The pipeline columns leave out cells crossing the antimeridian and the twelve pentagons, whose walls bend where they cross face edges, so that their corners joined by straight lines miss part of them; Kmoch et al. left out cells whose geometry their libraries drew invalid, which removes a few more. The last column is cell_metrics() on every cell, pentagons included, with areas on the sphere.

Cell by cell, the corners hexify draws are DGGRID’s for every ISEA3H, FULLER3H, ISEA7H and FULLER7H cell Kmoch et al. kept, and H3’s for every H3 cell, except the aperture-7 cells at odd resolutions whose DGGRID corners are misplaced (134 cells at resolution 3, 975 at resolution 5). On the rest, area and isoperimetric quotient agree with theirs to 0.03% and 2e-5.

Two readings follow. On the sphere, hexify’s ISEA hexagons of one resolution have exactly one area, and the spread in the last column comes from the pentagons alone; the spread of about 0.004 that both pipelines find for every ISEA grid on WGS84 is the latitude factor of the first section, the same for apertures 3, 4, 7 and 4/3. Aperture 3 and 4 hexagons, which Kmoch et al. left out as not congruent, have the same area and shape spread as aperture 7.

OGC API - DGGS

OGC API - Discrete Global Grid Systems (OGC 21-038r1) serves data organised by a Discrete Global Grid Reference System (DGGRS) and describes each DGGRS in a JSON schema. dggrs_definition() writes a hexify grid in that schema:

def <- dggrs_definition(g7)
str(def$dggh$definition)
#> List of 8
#>  $ spatialDimensions : int 2
#>  $ temporalDimensions: int 0
#>  $ crs               : chr "EPSG:4326"
#>  $ basePolyhedron    : chr "icosahedron"
#>  $ refinementRatio   : int 7
#>  $ refinementStrategy: chr [1:2] "centredChildCell" "nodeSharingChildCell"
#>  $ constraints       :List of 1
#>   ..$ cellEqualSized: logi TRUE
#>  $ zoneTypes         : chr [1:2] "hexagon" "pentagon"
def$zirs$textZIRS$description
#> [1] "Z7 (IGEO7): two digits naming the resolution-0 cell (00-11) the zone descends from, then one digit 0-6 per resolution naming the child of the parent: 0 the centred child, 1-6 the six around it; the string DGGRID writes. Dropping the last digit gives the parent's identifier."

The ISEA3H and ISEA7H that OGC registers are not hexify’s grids of those names: they convert geodetic latitude to authalic latitude and place the icosahedron’s first vertex at 11.20 degrees E and authalic latitude $\arctan(\phi)$ , where hexify, like DGGRID, reads latitude on the sphere and places it at 11.25 degrees E. Their zone identifiers therefore name other zones, and writing them waits on the authalic-latitude option (issue #104).

References

Gibb, R. (ed.) (2021). Topic 21: Discrete Global Grid Systems, Part 1: Core Reference System and Operations and Equal Area Earth Reference System. OGC Abstract Specification 20-040r3. https://docs.ogc.org/as/20-040r3/20-040r3.html

Griffin, B. (2026). Hex9: A Quasi-Authalic, Quasi-Continuous Hexagonal DGGS on the Reference Ellipsoid. arXiv:2608.00022.

Kmoch, A., Vasilyev, I., Virro, H. and Uuemaa, E. (2022). Area and shape distortions in open-source discrete global grid systems. Big Earth Data 6(3), 256-275. https://doi.org/10.1080/20964471.2022.2094926

Open Geospatial Consortium (2025). OGC API - Discrete Global Grid Systems - Part 1: Core. OGC 21-038r1. https://docs.ogc.org/is/21-038r1/21-038r1.html

Gilles Colling

2026-10-10