ISEA Projection

The Icosahedral Snyder Equal Area (ISEA) projection is what makes IGEO7 cells truly equal area. It maps the sphere onto the faces of an icosahedron such that area is perfectly preserved — no cell is larger or smaller than any other at the same resolution.

The Problem with Other Projections

Most map projections distort either area, shape, or both. For a DGGS, area distortion is the critical problem: if cells vary in size, then cell-count-based statistics are biased.

H3 uses the gnomonic projection (great circles become straight lines on each icosahedral face). This is fast and preserves straight-line paths, but cells near face edges are up to 50% smaller than cells near face centres.

IGEO7 uses ISEA, which eliminates this distortion entirely.

How ISEA Works

The ISEA projection is a two-step process:

Sphere → Icosahedron face: Each point on the sphere is mapped to the nearest icosahedral face using an equal-area azimuthal projection centred on that face. The Snyder equal-area projection ensures that the area of any region on the sphere equals the area of its projection on the face.
Face → 2D plane: The 20 triangular faces can then be unfolded into a flat map. IGEO7 uses a specific orientation (the ISEA orientation with vertex 0 at 11.2°E, see below) that minimises vertices on land.

The result: any two cells at the same resolution have exactly the same area.

Authalic Latitude

DGGRID implements ISEA on an authalic sphere. To use the grid on the WGS84 ellipsoid, geodetic latitudes are converted to authalic latitudes. The authalic latitude $\beta$ of a point at geodetic latitude $\phi$ on the WGS84 ellipsoid is defined as:

\sin\beta = \frac{q(\phi)}{q_p}

where $q(\phi)$ is the authalic helper function:

q(\phi) = (1 - e^2)\left(\frac{\sin\phi}{1 - e^2\sin^2\phi} - \frac{1}{2e}\ln\frac{1-e\sin\phi}{1+e\sin\phi}\right)

and $q_p = q(\pi/2)$ , $e$ is the WGS84 eccentricity ( $e \approx 0.0818$ ).

Authalic latitude preserves area relationships between parallels. Converting geodetic coordinates to authalic coordinates before the ISEA projection is applied ensures the equal-area property holds with respect to the real Earth shape, not just a sphere.

The conversion is explicit for now

The DGGRID tool currently does not apply the authalic conversion yet, which is why it still has to be done explicitly in dggrid4py: convert inputs with geoseries_to_authalic and outputs with geoseries_to_geodetic from dggrid4py.auxlat (see dggrid4py). Skipping it shifts latitudes by up to about 0.13 degrees, which puts points into the wrong cell. DGGRID v9 (already in beta) will have an option to do the conversion. DGGAL applies it automatically for ISEA7H_Z7.

IGEO7 Icosahedron Orientation

IGEO7 uses the ISEA orientation with vertex 0 at longitude 11.20, defined by:

dggs_vert0_lon      11.20
dggs_vert0_lat      58.28252559
dggs_vert0_azimuth  0.0

This places one icosahedron vertex near 58°N, 11°E (southern Scandinavia), which:

Avoids placing vertices on densely populated land
Maintains north–south symmetry for the grid

The latitude is an authalic latitude; in WGS84 that vertex is at 58.3971°N.

Set 11.20 explicitly with DGGRID v8

In the DGGRID v8 series, the IGEO7 type still uses ISEA7H's preset of dggs_vert0_lon 11.25, so 11.20 has to be set explicitly (see DGGRID). This will hopefully be fixed in the near future (DGGRID v9, already in beta). DGGAL's ISEA7H_Z7 uses 11.2 already.

Area Comparison: IGEO7 vs H3

At resolution 9, both IGEO7 and H3 produce cells of roughly "1 km scale". But their area distributions differ fundamentally:

Property	IGEO7 res 9	H3 res 9
Cell area	1.264 km² (all cells)	0.105 km² avg (varies ±50%)
Area variation	0%	up to ±50%
Equal area guaranteed	Yes	No

The ±50% figure for H3 is not a worst-case outlier — it is the systematic variation across all cells caused by the gnomonic projection distortion increasing toward icosahedral face edges.

For any analysis where cell count is used as a proxy for area — population density, land cover statistics, sampling intensity — this variation introduces bias. IGEO7 eliminates it.

Practical Implications

When equal area matters:

Density calculations (events per unit area)
Land cover and habitat area statistics
Climate model grid cells
Equal-weight spatial sampling
Regulatory reporting requiring area-accurate statistics

When equal area is less critical:

Nearest-neighbour lookups
Route finding / navigation
Visualisation-only applications

The Problem with Other Projections​

How ISEA Works​

Authalic Latitude​

IGEO7 Icosahedron Orientation​

Area Comparison: IGEO7 vs H3​

Practical Implications​

Further Reading​