Projection reference

Documentation index · Visual gallery · Forward/reverse API

Each projection family keeps its mathematical context, implementation contract, sources, and visual contact sheet together.

Equal Earth is currently a Stage 16J exploration-only comparison method. Its standalone forward/reverse API and five comparison plates do not add a seventh standard runtime or release projection.

Runtime API 3 implements forward and candidate-aware reverse support for all six families. Cuts, duplicated edges, poles, and Star-X carrier/cap overlap remain explicit candidates rather than being forced into a false unique answer. See the portable fixtures and independent reverse oracles.

Projection summaries

AuthaGraph

The AuthaGraph implementation follows Hajime Narukawa’s 2022 analytic formulation, orients the tetrahedron with the four published geographic vertices, and scales the unfolded periodic net to any valid map frame. A named A3 preset aligns projected coordinates with the checked-in AuthaGraph drawing sheet.

Construction examples, screen-coordinate conventions, optional raster naming, the exact frame predicate, and the source-plate ag_a3 preset are in the AuthaGraph public API notes.

Cahill-Keyes

The Cahill-Keyes implementation is derived from Mary Jo Graça and Gene Keyes’s MegamapMaker-prep9.pl, preserves the existing Visionscarto map registration, and scales to any finite, positive 2:1 a60::carto::frame.

Construction examples, compatibility presets, screen coordinates, and error behavior are in the Cahill-Keyes public API notes. The same guide distinguishes a valid 2:1 world carrier from arbitrary-ratio enlargement slices.

Star-X

Star-X reuses the native Cahill-Keyes geometry, splits the ordinary M layout into left and right groups of four spatial face slots, rotates the right group by 180 degrees, and stacks it above the left group. This produces the portrait X arrangement around the northern polar locus without raster tiles or temporary maps. Its default layout closes the former central gap, enlarges the complete X 120 percent about the page center, adds a central North-pole star, and composes Antarctic source geometry once at the lower end at the projection’s geographic scale. Natural Earth provides the standard physical content but does not control the cap geometry.

The current compositor cuts every source at the fixed 60°S parallel and reunites the four southern physical and graticule fragments around one bottom-center pole. It preserves ordinary source-tip radius and registers the complete projected boundary independently of any land dataset. Its bottommost point retains 0.25/44 of frame height—exactly 0.25 units on a 34-by-44 plate—so the blue edge remains inside the view box. Every unified Antarctic fragment is emitted after the ordinary quadrants and paints above them. The Star-X snapshot shows the released projection family; the next generated release will carry the corrected lower clearance.

Construction examples and star_x_layout configuration are in the Star-X public API notes. The signed gap and page-centered enlargement default to -9/88 of frame height and 1.2. The central star and unified Antarctica remain layer-aware SVG composition helpers, not hidden changes to the point transform; cap points preserve their ordinary source-tip radius while normalizing bearing so all four bent Cahill-Keyes edges meet. This fixed, data-independent cap makes a checked reverse practical: undo the carrier into Cahill–Keyes candidates, or recover cap bearing and solve latitude on [-90°,-60°], while retaining component identity for overlaps and cuts. The runtime still advertises Star-X reverse as candidates: component 0 owns the ordinary carrier north of 60°S, component 1 owns the cutoff and unified cap, and qualifiers resolve otherwise valid overlap candidates without guessing.

Dymaxion

The Dymaxion implementation uses Fuller’s geographic icosahedron orientation and Robert W. Gray’s exact sphere-to-equilateral-face equations. It places the result through the 23-piece horizontal Airocean net: 18 complete faces, two pieces of the Australia parent face, and three pieces of the Japan parent face. The local transform preserves uniform scale along every facet edge and is intentionally distinct from a radial, gnomonic projection.

The public frame may have any finite positive size that retains the exact 11/(3sqrt(3)) net ratio. Construction examples, formulas, face selection, split-face registration, generator behavior, and verification are in the Dymaxion implementation notes. The geometric context illustrates the facets, cuts, screen quadrants, graticules, and resulting Earth map.

Myriahedral

The Myriahedral implementation reproduces the depth-5 icosahedral mesh of temporaer/myriaworld and uses a fixed default tree reconstructed and registered for the checked-in source raster. Compact embedded minimum-spanning trees specify which of the 5120 small faces stay attached; the generation layer also includes five exploratory perspectives and two exact complementary face-group slices. The projection locates a spherical face, intersects its central ray with the face plane, transfers the gnomonic barycentric coordinates to the unfolded planar copy, and scales the complete net to a variable frame. It has no runtime dependency on the historical generator, Boost.Graph, S2, GDAL, or Natural Earth.

Construction examples, the myriahedral_source preset, and the distinction between this raster-registered 16:9 net and other possible Myriahedral nets are in the Myriahedral public API notes. The same notes record the perspective configurations and Myriahedral slicing. The production WebAssembly base-map option emits only the ocean and land layers.

Icosahedral Voronoi

The Voronoi implementation reproduces the default geoIcosahedral() layout from d3-geo-polygon without a JavaScript dependency. Twelve spherical vertices define twenty regular triangular faces. Each triangle centroid is a Voronoi site; the forward transform selects the nearest site by maximizing its dot product with the geographic unit vector, projects onto that face’s gnomonic tangent plane, and carries the result through the fixed shared-edge unfolding tree.

This is distinct from the Myriahedral projection above: both begin with an icosahedron, but this projection retains 20 regular faces and a conventional fixed net, while Myriahedral subdivides to 5120 faces and uses a land-aware tree.

Construction examples, source-canvas registration, screen coordinates, and the voronoi_source preset are in the Voronoi public API notes.