Equal Earth implementation

Context · Sources · Portable fixtures · Stage 16J comparisons

Stage 16J supplies the spherical Equal Earth forward and reverse equations in the standalone C++ header cart0freak0-equal-earth.h and the dependency-free JavaScript module equal-earth.mjs. Both expose raw radian-space operations and full-carrier, top-left normalized page mapping. The C++ page wrapper returns unique or outside; this uninterrupted method does not need the face-qualified candidate sets used by interrupted Cartofreako carriers.

The implementation uses the published coefficients:

A1 =  1.340264
A2 = -0.081106
A3 =  0.000893
A4 =  0.003796
M  = sqrt(3) / 2

For longitude lambda, latitude phi, and theta = asin(M sin(phi)), the forward equations are:

x = lambda cos(theta) /
    (M (A1 + 3 A2 theta² + theta⁶ (7 A3 + 9 A4 theta²)))
y = theta (A1 + A2 theta² + theta⁶ (A3 + A4 theta²))

Reverse projection solves theta from y with at most 12 Newton steps, then reconstructs longitude and latitude. Unit-domain clamps apply only at the inverse trigonometric boundary. A page point is accepted only when its reconstructed longitude remains within the selected antimeridian and its forward residual returns to the requested page point. Page corners outside the curved carrier are therefore outside, not silently clamped to land. The two page edges are duplicate representations of the selected antimeridian. Reverse projection accepts both and reports the same canonical longitude; its residual check recognizes the equivalent opposite edge without relaxing interior-point validation.

Checked numerical evidence

Run:

make check-equal-earth-projection

The offline gate validates 30 neutral cases in two layouts through C++ and JavaScript. The frozen cross-implementation observations use PROJ 9.6.2 and D3 Geo 2.0.1. On the development run:

• the maximum PROJ-to-D3 raw-coordinate difference was 7.021666937153402e-16;
• the maximum JavaScript-to-fixture raw difference was the same;
• the maximum forward/reverse angular difference was 2.842170943040401e-14°; and
• the maximum finite-difference spherical area-scale error over 35 diagnostic samples was 1.203175781228083e-9.

These are implementation diagnostics, not a claim about survey accuracy, ellipsoidal geodesy, campaign compliance, or human perception. The diagnostic report is generated locally at build/stage-16j/equal-earth-diagnostics.json.

Runtime boundary

Equal Earth intentionally remains outside projection_kind, runtime API 3, the standard artifact manifest, make all, print generation, GitHub release, and UCB AAO/S3 publication. Stage 16J uses it through a dedicated API and comparison renderer so implementing a research control cannot mutate the six-family release corpus by accident. Promotion would require a separate runtime/API, pass-matrix, gallery, print, and lifecycle decision.