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 standard ISEA orientation) that minimises vertices on land.

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

Authalic Latitude

DGGRID implements ISEA using authalic latitudes internally. 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. DGGRID converts geodetic coordinates to authalic coordinates before applying the ISEA projection, ensuring the equal-area property holds with respect to the real Earth shape, not just a sphere.

Interoperability

When working with WGS84 lat/lng coordinates (GPS, GeoJSON), DGGRID handles the authalic conversion automatically. You pass standard geographic coordinates in and get Z7 cell IDs out. The conversion is internal.

IGEO7 Icosahedron Orientation

IGEO7 uses the standard ISEA orientation, defined by:

dggs_vert0_lon     11.25
dggs_vert0_lat     58.2825255885
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
Is the same orientation used by DGGAL and dggrid4py by default

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​