A · canonical continuous / 2 km
The 21,600 × 10,800 January, May, and November source is reduced with CONV before the canonical continuous inverse reads its working lattice. This is the original 55,167-support map, preserved byte-for-byte.
Five-Foot World
Let exact hexagons make imperceptible five-foot registry concessions, repeatedly and implicitly, until the poles open into one continuous, Eckert-like Earth.
Drag or use two fingers to pan. Pinch to zoom exactly toward the point under the pointer. Double-click to inspect a place. Press 0 to return to the whole Earth.
A is the canonical continuous projection from the nominal 2 km source: CONV reduces the source before the established centroid-local inverse evaluates the 8K map.
How the surface is flattened
The continuous construction begins with the p=0.5 member of a Mollweide–Eckert family: the equal-area geometry of Eckert IV, normalized so each polar line is half the width of the equator. A flat-top hexagonal support lattice covers that wider footprint. The extra polar width admits 6,640 more supports than the former Mollweide oval. Every projected point consults a compact neighborhood of centroid-local tangent inverses, blends them as spherical unit vectors, and then asks the Earth texture for the resulting longitude and latitude.
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Resolve the local direction
Each point tests a fixed 31-centroid candidate stencil. The compact Wendland C2 kernel makes no more than 19 supports active at once, reaches exactly zero at its boundary, and normalizes the surviving weights.
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Share the stress
The analytic equal-area Jacobian supplies a distortion potential at each centroid. Sixty-four Jacobi passes interpolate that field across the six-neighbor graph; the result is scaled so the largest adjacent centroid difference is exactly 1.524 metres. There are no rendered slip fronts: the renderer evaluates the final continuous registry directly, so the many small concessions remain implicit.
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Hold the geometry fixed
A and B evaluate the identical p=0.5 base, identical 55,167 centroids, and identical 35:65 analytic-to-local inverse. A begins with the nominal 2 km source; B instead reconstructs directly from the lossless 86,400 × 43,200 source into each final 8K pixel.
These are two source-and-reconstruction renders of one projection. Geometry is controlled: only source support and reconstruction change. B has no projected nearest-neighbor master, no bilinear sampler, and no lossy intermediate.
One projection, two source supports
A remains the established continuous projection and its nominal 2 km source. B replaces the former geometric experiment with the same continuous projection rebuilt from the lossless 500 m tiles. Both occupy the same 8,192 × 4,608 frame, so their visible difference can now be attributed to source support and reconstruction rather than to a moving map.
B · canonical continuous / 500 m
The lossless-Deflate 86,400 × 43,200 tiles are composed at native-cell resolution and decoded to linear RGB. Sixteen Gauss–Legendre sites in every final pixel query the canonical inverse, then a bounded paired CONV reconstruction reads the native source directly.
A is the existing 3,919,767-byte delivery image at its recorded 41.76 dB result. B is a 22,867,719-byte baseline quality-100, 4:4:4 encode made directly from its 452,984,832-byte linear floating-point master. The larger B payload is deliberate: it preserves the direct reconstruction without passing through the ringing-producing optimization experiments.
What the numbers actually say
The p=0.5 base is exactly equal-area. The final map is not: its centroid correction deliberately spends some of that exactness to negotiate directional stress without cuts. The audit below measures the rendered geometry, not the appearance of the satellite texture.
The important distinction is between integrated and local error. Greenland’s expansions and contractions partly cancel over the island, producing a +0.4921% whole-island result, while its sampled local scale runs from −10.42% to +6.81% with a +0.284% median. Globally, the median sampled area error is +0.00781%. Shape is the larger sacrifice: the 95th-percentile angular-axis ratio is 13.74 in this deliberately polar-stretched version. The poles themselves are excluded from the numerical grid because longitude is singular there.
Where this version lives on the Pareto surface
There is no single scalar called projection accuracy. Modern experimental projections choose a loss function and a viewing purpose, then negotiate among infinitesimal area, angular shape, distance, finite-feature bending, boundary form, interruptions, invertibility, and computational cost. The criteria below measure the canonical continuous field shared exactly by A and B.
- Areal fidelity
A = √Ecos φ[(ln sA)²]. Zero is exact equal-area. This final field hasA = 0.014951; its base has zero.- Angular shape
K = σmax/σminandω = 2 asin((K−1)/(K+1)). The final area-weighted RMSωis 34.9998°.- Rectangular occupation
F(p) = p + (1−p)π/4. Atp = 0.5, the map occupies 89.2699% of its 2:1 bounding rectangle and its polar half-width is 50% of the equator’s.- Continuity
- Zero interruptions, zero empty interior pixels, and a maximum numerical forward/inverse residual of 1.15 × 10−15 radians.
- Discrete burden
- 55,167 unchanged regular-hex supports—6,640 more than the Mollweide endpoint—while at most 19 compact supports influence a pixel.
- Implicit slack
max |hi−hj| = 1.524 mon neighboring centroids, 0.000940° maximum modeled tilt, and no explicit slip event or slip front in the raster.
The centroid correction lowers K95 from 14.2687 to 13.7355—a 3.74% improvement—and moves 95% of sampled points no more than 3.102 pixels from the exact p=0.5 base. It pays with nonzero area error, a nearly unchanged angular RMS (34.9724° to 34.9998°), and a larger worst sampled angular ratio (16.6564 to 28.9803).
So this version is not a universal replacement for Eckert IV. It is non-dominated only when the brief values the smaller central angular tail and an exact, bounded hex-and-slip realization alongside classical cartographic error. For strict area analysis, its own p=0.5 base remains better. For the requested continuous, more rectangular visual world, this is the measured location of the compromise.
B changes the sampled field, not the map
For every final pixel, B places a 4 × 4 Gauss–Legendre quadrature over that pixel’s target-space footprint. Each of its sixteen sites passes through the same continuous inverse used by A. The resulting longitude and latitude are evaluated against the native 500 m seasonal field by the paired arbitrary-site BFFT CONV interpolant:
CB[p] ≈ Σi=1…4 Σj=1…4 wiwj 𝒞500m(TA−1(xij, yij))
The quadrature integrates the warped source over the output pixel instead of asking one point to stand in for the whole footprint. CONV is evaluated horizontally and vertically with bounded 11-row support in linear RGB. Projection coordinates and quadrature remain IEEE-754 binary64; the authoritative production CONV backend evaluates its local reconstruction in binary32.
| Controlled term | A · continuous / 2 km | B · continuous / 500 m |
|---|---|---|
| Projection | Canonical p=0.5 continuous centroid-local inverse | Identical |
| Source | Nominal 21,600 × 10,800 global JPEGs | Lossless native 86,400 × 43,200 tiled source |
| Reconstruction | CONV reduction before final inverse evaluation | Direct paired arbitrary-site CONV at sixteen sites in every final footprint |
| Intermediate | Established A working lattice | No projected raster; linear floating-point accumulation only |
| Delivery | 3,919,767 bytes; recorded 41.76 dB optimization | 22,867,719 bytes; quality 100 and 4:4:4, encoded once |
B’s linear master is 452,984,832 bytes. Its only range repair is a neutral RGB shift on 1,871 pixels, with no channel clipping and no unrepairable pixel; the largest shift is 0.026575 in linear-light units. The final JPEG’s decoded RMS deviation from the prepared 8-bit carrier is 0.734 code value. Those are encoding facts, not projection improvements: A and B have exactly the same area, angular, distance, boundary, and continuity behavior.
How it compares with maps that look plausible at world scale
Exact equal-area projections still win the area contest: zero theoretical area error is better than a small measured one. Five-Foot World is interesting for a different bargain—a more rectangular unbroken world, a smooth centroid-controlled deformation field, and a physically bounded but visually implicit slack variable rather than cuts or visible facets.
| Approach | Relative area | Where distortion goes | Visible map |
|---|---|---|---|
| Five-Foot continuous · A and B | Exactly equal-area before the centroid correction; not exact afterward. Greenland: +0.4921%. | Compact centroid blends, plus a hidden height-and-tilt field capped at five feet between neighbors. | One continuous Eckert-like world; no interior holes, slip fronts, or drawn tile boundaries. A and B differ only in source support and reconstruction. |
| Mollweide | Exactly equal-area. | Shape and angle, especially toward the rim and high latitudes. | One continuous oval. |
| Equal Earth | Exactly equal-area. | Shape and angle, tuned for a familiar continental appearance. | One continuous pseudo-cylindrical world. |
| Eckert IV | Exactly equal-area. | Angular and shape deformation within a rounded footprint. | One continuous rounded world. |
| Interrupted Goode Homolosine | Exactly equal-area. | Multiple interruptions prevent distortion from accumulating across oceans. | Better continental shapes, but conspicuous cuts. |
| Icosahedral Snyder Equal Area | Exactly equal-area on its polyhedral construction. | Distortion is distributed over faces and their topology. | Faceted layouts or unavoidable cuts when unfolded. |
| Winkel Tripel / Robinson | Compromise projections; neither is exactly equal-area. | Area, angle, distance, and shape all share the error. | Familiar, continuous atlas-style worlds. |
For strict quantitative area work, use Mollweide, Equal Earth, Eckert IV, Goode, or ISEA. For a continuous visual world in which deformation is locally negotiated and the residual has a physical interpretation, this construction offers a distinct and measurable alternative.
Sources and reproducibility
Both images derive from NASA Blue Marble: Next Generation with Topography and Bathymetry for January, May, and November. A uses NASA’s nominal 2 km global JPEGs. B uses the lossless-Deflate 500 m tiled GeoTIFFs, direct target-footprint CONV reconstruction in linear RGB, neutral gamut shift, and one baseline quality-100, 4:4:4 JPEG encode. Both use the same canonical continuous projection. These are data-derived monthly composites, not single camera exposures. The projection criteria are grounded in the primary papers for Natural Earth II and Equal Earth; definitions and comparison properties follow the primary documentation for Equal Earth, Eckert IV, Interrupted Goode Homolosine, ISEA, and Winkel Tripel.