Equirectangular Projection¶
A cylindrical map family that sends spherical longitude and latitude to separate linear plane coordinates, with true east-west scale at chosen standard parallels.
Core Idea¶
The equirectangular or equidistant cylindrical projection maps a sphere's longitude and latitude separately to horizontal and vertical plane coordinates. With sphere radius \(R\), central longitude \(\lambda_0\), vertical origin \(\phi_0\) and standard-parallel latitude \(\phi_s\), its spherical equations are \(x=R(\lambda-\lambda_0)\cos\phi_s\) and \(y=R(\phi-\phi_0)\) for angles in radians. Meridians and parallels form an evenly spaced rectangular grid. The \(\phi_s=0\) member is plate carrée; \(\phi_s=45^\circ\) is Gall isographic, the narrower discovery candidate reframed into this family.[ref-6ab22d7269b0][ref-b7ad4af4eb60]
Scope of Application¶
The simple grid supports thematic maps, globally indexed rasters and planetary map images. NASA's Planetary Data System defines an Equirectangular projection class with origin and standard-parallel parameters. Spherical-video metadata can also identify equirectangular storage of viewing directions in a rectangular frame. These are uses of angular indexing, not guarantees that equally sized image pixels cover equal ground or solid angle. A true ellipsoidal map needs different formulas, including meridional arc in the vertical coordinate.[ref-6ab22d7269b0][ref-b7ad4af4eb60][^ref-76e6f580039e]
Clarity¶
The “equidistant” name is limited: meridian distances are true, and local east–west scale is true at \(\pm\phi_s\). At latitude \(\phi\) the spherical horizontal scale is \(\cos\phi_s/\cos\phi\), not universally \(1/\cos\phi\); that simpler ratio holds only for plate carrée. For Gall isographic, scale is compressed to \(\sqrt2/2\) at the equator, true at \(\pm45^\circ\), and stretched to \(\sqrt2\) at \(60^\circ\). It is not an equal-area or conformal world map.[ref-6ab22d7269b0][ref-a846d0b7f5f5]
Manages Complexity¶
The separable equations allow direct coordinate lookup and inverse lookup: away from the seam and poles, \(\lambda=\lambda_0+x/(R\cos\phi_s)\) and \(\phi=\phi_0+y/R\). This makes regular angular images easy to index. The tradeoff is that a rectangle of constant pixel size represents different surface areas as latitude changes; the poles stretch into map lines and a longitude seam must be declared. Any area or distance analysis must restore the sphere's metric rather than treating pixel geometry as ground geometry.[ref-6ab22d7269b0][ref-b7ad4af4eb60]
Abstract Reasoning¶
Choose \(R\), an angular origin, a standard parallel and a longitude seam. Check that \(x\) is linear only in longitude and \(y\) linear only in latitude for the spherical model. Compare map differentials with ground differentials: \(dy/(R\,d\phi)=1\) and \(dx/(R\cos\phi\,d\lambda)=\cos\phi_s/\cos\phi\). These ratios identify where scale is true and why latitude-dependent area distortion occurs. On an ellipsoid, replace the spherical rule with the declared ellipsoidal construction; do not carry over linearity in geodetic latitude.[^ref-6ab22d7269b0]
Knowledge Transfer¶
The angular-grid pattern can be instantiated on Earth, another spherical body or a viewing sphere when longitude/latitude interpretation and standard-parallel scaling are preserved. Plate carrée and Gall isographic differ in \(\phi_s\), not in the family rule. A rectangular photograph without spherical-angle coordinates, or a Mercator map with nonlinear latitude spacing, is not the same projection. The portable insight is simple angular indexing with explicit, latitude-dependent metric distortion.[ref-6ab22d7269b0][ref-b7ad4af4eb60][^ref-76e6f580039e]
[^ref-6ab22d7269b0]: PROJ contributors, “Equidistant Cylindrical (Plate Carrée)”, primary operation documentation, special-cases table and spherical/ellipsoidal formulas. Its contradictory “Conformal cylindrical” table cell is not adopted. [^ref-b7ad4af4eb60]: NASA Planetary Data System, “Equirectangular” cartography data-dictionary class, Description and parameters. [^ref-a846d0b7f5f5]: Esri, “Equidistant cylindrical” ArcGIS Pro documentation, Projection properties and Variants. [^ref-76e6f580039e]: Google Spatial Media, Spherical Video V2 RFC, equirectangular projection mode.
Relationships to Other Abstractions¶
Current abstraction Equirectangular Projection Domain-specific
Parents (1) — more general patterns this builds on
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Equirectangular Projection is a kind of Projection Prime
An equirectangular map is a particular sphere-to-plane projection with a fixed angular rule and standard-parallel scale behavior.
Hierarchy path (1) — routes to 1 parentless root
- Equirectangular Projection → Projection → Abstraction
Neighborhood in Abstraction Space¶
Equirectangular Projection sits in a moderately populated region (53rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Albers Equal-Area Conic Projection — 0.91
- Gnomonic Projection — 0.89
- Vincenty's formulae — 0.86
- Rhumb line — 0.85
- Haversine Formula — 0.84
Computed from structural-signature embeddings · 2026-10-08