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.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13198
Domain group: Natural Sciences
Origin domain: Geology & Earth Sciences
Subdomains: Cartographic Projection Design, Spherical Coordinate Mapping → Geology & Earth Sciences
Aliases: Equidistant Cylindrical Projection, Simple Cylindrical Projection, Rectangular Projection

Core Idea¶

The equirectangular projection, also called the equidistant cylindrical family, converts angular position on a sphere into rectangular plane coordinates by scaling longitude and latitude separately. For spherical radius \(R\), central meridian \(\lambda_0\), latitude of origin \(\phi_0\), and chosen standard-parallel magnitude \(\phi_s\), its canonical forward rule (angles in radians) is

\[ x=R(\lambda-\lambda_0)\cos\phi_s,\qquad y=R(\phi-\phi_0). \]

False eastings/northings may translate the grid without changing the family. Meridians therefore become equally spaced vertical lines and parallels equally spaced horizontal lines. The standard parallel is the family parameter: \(\phi_s=0\) produces plate carrée; \(\phi_s=45^\circ\) produces the Gall isographic member that led to this reframe. The grid is computationally simple, but equal spacing in the map does not mean equal ground area or shape.^[1]^[2]

On the sphere, a north–south latitude increment has ground length \(R\,d\phi\) and map length \(dy=R\,d\phi\), so meridian scale is true. An east–west longitude increment at latitude \(\phi\) has ground length \(R\cos\phi\,d\lambda\) and map length \(dx=R\cos\phi_s\,d\lambda\). Its local scale is therefore \(k_E=\cos\phi_s/\cos\phi\). It equals one at \(\pm\phi_s\), is compressed at the equator when \(\phi_s\ne0\), and stretches increasingly poleward beyond the standard parallels. For plate carrée alone, this becomes \(1/\cos\phi\). The simple angular formulas are spherical; the true ellipsoidal version uses meridional arc rather than \(R\phi\) and must be declared separately.^[1]^[2]

Structural Signature¶

Sig role-phrases: spherical angular source → longitude/latitude reference origin → standard parallel → separate linear angle-to-plane rule → rectangular graticule with a determinate distortion field.

Angular source and body model. A location is supplied as longitude \(\lambda\) and latitude \(\phi\) on a sphere of radius \(R\). A planet, camera-viewing sphere or spherical Earth approximation may supply that surface. Merely drawing an arbitrary rectangular image does not identify this projection. A geodetic ellipsoid requires the separately defined ellipsoidal equidistant cylindrical equations.^[1]^[2]
Reference meridian and origin. \(\lambda_0\) fixes the grid's center and longitude seam; \(\phi_0\) fixes the vertical origin. Changing these translates or recenters the map while retaining the angle-scaling family. Longitudes must be unwrapped consistently across a chosen seam, or an apparent jump in \(x\) is unavoidable.
Standard parallel. \(\phi_s\) fixes the longitude multiplier \(R\cos\phi_s\). The same factor yields true east–west local scale at northern and southern latitudes \(\pm\phi_s\). The choice changes the distortion profile, not just a label: plate carrée has \(0^\circ\) and Gall isographic \(45^\circ\).^[1]
Separable rule. Horizontal position is linear in longitude difference and vertical position linear in latitude difference on the sphere. This is the defining map operation. A projection with a nonlinear latitude formula, such as Mercator, is not a member merely because its meridians and parallels are straight.
Rectangular outcome and metric boundary. Equal angular increments make equal planar intervals and rectangular grid cells. North–south scale is true along meridians, but east–west scale varies as \(\cos\phi_s/\cos\phi\) and tends to infinity at the poles. The map is neither globally equal-area nor conformal; a single pole occupies a full horizontal line of map coordinates unless a seam convention collapses it.^[1]^[3]

What It Is Not¶

It is not plate carrée only. Plate carrée is the \(\phi_s=0\) member; Gall isographic, with \(\phi_s=45^\circ\), is another. Calling any lat–lon raster “plate carrée” can hide the chosen standard parallel and therefore its scale. It is not Gall isographic only: that named candidate has been reframed upward to the whole parameterized equirectangular family, without erasing the 45-degree case.^[1]

It is not equal-area, conformal, or a map that preserves every distance. A small longitude interval is stretched or compressed by \(k_E\) except on the standard parallels, while meridian distance remains true. Away from standard parallels the unequal north–south/east–west scales alter local shape; varying area scale rules out equal-area status. The adjective “equidistant” is qualified by direction, not a license to measure arbitrary point-to-point distances with a ruler.^[3]

It is not Mercator: Mercator changes vertical spacing nonlinearly to preserve local angles, whereas equirectangular keeps latitude spacing linear in the spherical case. It is not Hammer or Aitoff, which use different formulas and nonrectangular whole-world outlines. It is not a physical pixel grid alone: the same rectangular image can encode camera frames, spreadsheet cells or a nongeographic matrix without any spherical-angle meaning. Nor does the simple spherical formula silently become exact when geographic coordinates refer to an ellipsoid.^[1]

Scope of Application¶

The family is suited to displaying and indexing angularly gridded data. Because \(x\) and \(y\) follow longitude and latitude with independent scale factors, locating a geographic coordinate in a plate-carrée raster is a direct affine lookup after choosing the central meridian, origin and image dimensions. PROJ documents thematic mapping and global rasters as common uses, and cautions against treating the projection as suitable for navigation or cadastral measurement where its distortion matters.^[1]

Planetary mapping is a literal application: NASA's Planetary Data System defines an Equirectangular cartography class and records parameters including projection origin and standard parallel. The mapped object need not be Earth; it must, however, have a declared spherical body/radius for the simple formulas or an appropriately declared extension. A raster pixel's regular row and column position does not certify equal surface area on the planet.^[2]

Spherical video and panoramic imaging use an angular rectangle similarly. Google's Spatial Media V2 specification contains an equirectangular projection mode for spherical video frames. In a full 360-degree view, horizontal raster position represents viewing azimuth and vertical position elevation or latitude, subject to image-axis conventions. This is angular storage, not a claim that video pixels have equal solid angle or that every cartographic standard-parallel choice appears in consumer video. The usual full-sphere frame is analogous to the \(\phi_s=0\) angle grid.^[4]

Clarity¶

State the body model, angular units, \(\lambda_0\), \(\phi_0\), \(\phi_s\), and any false easting/northing before interpreting coordinates. On a unit sphere with all origins zero and \(\phi_s=0\), a location \(30^\circ\) east and \(60^\circ\) north maps to \((\pi/6,\pi/3)\) in radius units. At that latitude its east–west map scale is \(1/\cos60^\circ=2\), while its meridian scale remains one. An apparently square grid cell at \(60^\circ\) thus represents less spherical east–west ground distance than an equally wide equatorial cell.^[1]

For Gall isographic, setting \(\phi_s=45^\circ\) changes \(x\) to \((\sqrt2/2)R(\lambda-\lambda_0)\). The east–west scale is \(\sqrt2/2\) at the equator (compression), one at \(45^\circ\) north or south, and \(\sqrt2\) at \(60^\circ\) (stretch). The slogan “east–west distortion grows like \(1/\cos\phi\)” is therefore incomplete for this member: the correct multiplier is \(\cos\phi_s/\cos\phi\). The named standard is not just a cartographic preference; it changes a measurable distortion field.^[1]

Manages Complexity¶

The mapping reduces spherical location lookup to two independent arithmetic operations, making inverse lookup equally simple away from conventions at the seam and poles. In the spherical form, \(\lambda=\lambda_0+x/(R\cos\phi_s)\) and \(\phi=\phi_0+y/R\). This structure makes regular angular rasters easy to store, tile, label and index; a data pipeline can translate an image row and column back to geographic angles without solving a nonlinear inverse. That convenience is why the rectangular form survives in thematic maps and spherical-media storage.^[1]^[4]

The apparent simplicity can conceal geometry. Equal angular grid cells change physical area with latitude, and the pole is represented by many horizontal positions referring to the same location. A workflow that totals raw pixels as if each covered equal ground area misuses the representation. If area comparison, shape fidelity or short-route navigation is central, another projection or an explicit area-weighting method may be required. The equirectangular map manages coordinate complexity by shifting responsibility for distortion analysis to the user.^[3]^[1]

Abstract Reasoning¶

To recognize the family, examine how coordinates change under small independent angular increments. A fixed longitude increment should give the same \(\Delta x\) at every latitude; a fixed latitude increment should give the same \(\Delta y\) everywhere on the sphere. Then identify the horizontal coefficient and ask whether it equals \(R\cos\phi_s\) for a declared \(\phi_s\). A merely rectangular graticule is not a sufficient test if the vertical latitude spacing or model differs. On an ellipsoid, the exact vertical coordinate becomes meridional arc \(M(\phi)-M(\phi_0)\); calling that “linear in geodetic latitude” would be false.^[1]

Distortion follows directly from a comparison of infinitesimal map and ground lengths. North–south, \(dy/(R\,d\phi)=1\). East–west, \(dx/(R\cos\phi\,d\lambda)=\cos\phi_s/\cos\phi\). This derivation identifies both standard parallels and the polar problem without relying on a visual impression of a world map. Because the two principal scales are unequal except at the standards, local angles and shapes are generally distorted; because their product varies with latitude, area is not generally preserved. These inferences are local; they do not make arbitrary distances between two distant points true.^[1]^[3]

Knowledge Transfer¶

The same angular-to-rectangular structure can transfer from terrestrial thematic maps to another spherical body or a 360-degree viewing sphere. Preserve the interpretation of both angular axes, the standard-parallel coefficient or equivalent horizontal scaling, the chosen origin, and the seam convention. A coordinate grid that looks identical on paper is not necessarily the same projection if one application treats its vertical coordinate as a nonlinear meridional arc on an ellipsoid or as an unrelated image pixel index.^[2]^[4]

The core computational insight is that independent angular coordinates become independent planar indices. The accompanying warning transfers too: a regular raster is not a regular metric tessellation of the sphere. A planetary scientist calculating mapped area and a panorama algorithm comparing pixel coverage must each account for latitude-dependent geometry rather than importing equal-pixel-area assumptions from the rectangle.

Examples¶

Canonical: plate carrée on a unit sphere¶

Set \(R=1\), \(\lambda_0=\phi_0=0\) and \(\phi_s=0\). Then \(x=\lambda\) and \(y=\phi\) with angles in radians. The point \((\lambda,\phi)=(30^\circ,60^\circ)\) maps to \((\pi/6,\pi/3)\). Meridian distance is preserved, but a small east–west segment at \(60^\circ\) is drawn twice its ground length. The entire globe occupies a \(2\pi\)-by-\(\pi\) angular rectangle under a conventional longitude interval, with a seam at its left/right edges. The pole maps to a horizontal edge even though all longitudes meet at one spherical point.^[1]^[2]

Mapped back: unit spherical angular source → zero reference meridian and origin → equatorial standard parallel → independent \(x=\lambda,y=\phi\) rule → rectangular graticule, meridian scale one and east–west scale two at \(60^\circ\).

Reframed candidate: Gall isographic¶

Keep the unit sphere and zero origins, but set \(\phi_s=45^\circ\). The longitude multiplier becomes \(\cos45^\circ=\sqrt2/2\), while \(y=\phi\) stays unchanged. The two standard parallels \(\pm45^\circ\) have true east–west scale; at the equator the horizontal scale is \(\sqrt2/2\), and at \(60^\circ\) it is \(\sqrt2\). This is a different member of the same parameterized family, not a new rule for vertical spacing or an equal-area map.^[1]

Mapped back: unit spherical source → zero origin → Gall's \(45^\circ\) standard-parallel choice → \(x=(\sqrt2/2)\lambda,y=\phi\) → identical rectangular graticule type but shifted distribution of east–west scale.

Applied: planetary equirectangular image grid¶

NASA's Planetary Data System records an Equirectangular cartography class with a latitude of projection origin and standard-parallel parameter. A planetary raster can therefore associate each row and column with an angular location on the mapped body using declared projection parameters. The archive description supplies the coordinate operation; it does not make equal-sized image pixels equal-area surface patches. The same distinction matters when counting pixels as observations or when measuring a feature's area.^[2]

Mapped back: declared planetary sphere/radius → recorded origin → recorded standard parallel → NASA PDS angular forward/inverse rule → regular raster with latitude-dependent ground-area scale.

Structural Tensions¶

Simple indexing versus faithful metric reading. Linear angle axes and rectangular pixels simplify storage, lookup and resampling. They also invite the false inference that equal-width pixels cover equal physical lengths or areas. A projection preserving area would improve area comparison but sacrifice this specific separable angular grid. Diagnostic: Is the immediate job angular address lookup and display, or accurate comparison of distance, shape and area across latitudes?^[1]^[3]

Center a region's east–west scale versus shift error elsewhere. Raising \(\phi_s\) moves true parallel scale away from the equator toward the chosen latitude band. That may help a midlatitude region, but it compresses the equator and stretches latitudes poleward of the standards. Choosing \(\phi_s=0\) gives true equatorial scale but larger east–west stretch at a midlatitude region. Diagnostic: Which latitude band bears the measurement burden, and how much scale error can other bands tolerate?^[1]

Structural–Framed Character¶

The coordinate law is chiefly structural. Its evaluative weight is low: the same formula can be appropriate for a raster index and inappropriate for an area-comparative map; the formula itself does not choose the task. Human practice matters in selecting central meridian, seam and standard parallel, and in deciding how much distortion is acceptable, but those choices parameterize rather than constitute the mathematical rule. The projection's historical and institutional names are not proof of its geometry; plate carrée, Gall isographic and modern EPSG/PROJ variants are recognizable by parameters and equations. Vocabulary travels from cartography to planetary mapping and spherical video because the same angular rectangularization can be instantiated. That travel is recognition of a shared transform, not an assertion that a video pixel is a parcel of land. The method is imported into applications as a coordinate format, while its distortion can be derived independently from the sphere's metric. Its character: a structural cartographic transformation with a task-sensitive frame for interpreting its output.^[1]^[2]^[4]

Structural Core vs. Domain Accent¶

The transferable skeleton is separate linear indexing of two angular coordinates into a rectangle. The live Projection prime already supplies the broader mapping-to-representation idea, so this entry does not propose a new prime. The domain-specific accent is indispensable: angular longitude/latitude on a spherical body, the standard-parallel coefficient, and the exact meridian/parallel scale law. If one removes the spherical metric, an arbitrary image-coordinate transform may still be linear but is no longer the equirectangular cartographic identity.^[1]

There is an ellipsoidal extension, but it must be named. PROJ's ellipsoidal equidistant cylindrical operation uses \(x=\nu_s\cos\phi_s(\lambda-\lambda_0)\) and \(y=M(\phi)-M(\phi_0)\), with \(M\) the meridional arc and \(\nu_s\) the prime-vertical curvature radius at the standard parallel. It preserves the analogous meridional distance property while abandoning the spherical claim that \(y\) is linear in geodetic latitude. Treating the simple formula as universally exact would erase a genuine body-model boundary.^[1]

This entry is a kind of Projection.

The broader abstraction is live Projection. The encyclopedia's prime explicitly includes flat maps of curved surfaces as instances of a broader move from richer source geometry to a selected representation. Equirectangular projection adds a specific angular rule and scale choice. The prime's idempotence idiom should not be applied as literal function composition here: a sphere-to-plane function cannot simply be re-applied to its planar output. The sound genus claim is representational projection, not a type-invalid algebraic equation.^[1]

The live Hammer Projection, Aitoff Projection and Mercator variants are neighboring map constructions, not strict parents: their forward formulas and distortion guarantees differ. In particular Hammer prioritizes area preservation and an elliptical outline, while equirectangular prioritizes separable angular coordinates and a rectangular outline. Related ideas such as regular grids or coordinate transforms explain parts of the mechanism but do not replace its defining map rule.

Relationships to Other Abstractions¶

Current abstraction Equirectangular Projection Domain-specific

Parents (1) — more general patterns this builds on

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 (53^rd 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

Not to Be Confused With¶

Plate carrée: the \(\phi_s=0\) member, not the entire equirectangular family.
Gall isographic: the \(\phi_s=45^\circ\) member, formerly the seed's narrower title.
Mercator: straight meridians and parallels but a nonlinear vertical mapping designed for conformality.
Hammer/Aitoff: different whole-world formulas and nonrectangular outlines; Hammer is equal-area.
Equidistant as all-pairs distance preservation: only meridian lengths and local lengths along the standard parallels are true under the stated spherical model.
Equal-area raster: equal planar pixel area does not imply equal spherical surface area.
Ellipsoidal exactness from spherical equations: true ellipsoidal coordinates use meridional arc and ellipsoid curvature terms.^[1]^[3]

References¶

[1] PROJ contributors, “Equidistant Cylindrical (Plate Carrée)”, primary operation documentation, Usage, special-cases table, Spherical formulas and Ellipsoidal formulas. The page's isolated “Conformal cylindrical” classification table cell conflicts with its equations and is not used. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w

[2] NASA Planetary Data System, “Equirectangular” cartography data-dictionary class, Description and projection parameters; cites Snyder (1987) p.90. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[3] Esri, “Equidistant cylindrical” ArcGIS Pro documentation, Projection properties and Variants, primary software-provider account of graticule, scale and ellipsoid caveat. registry ↩a ↩b ↩c ↩d ↩e ↩f

[4] Google Spatial Media, Spherical Video V2 RFC, projection-data and equirectangular mode sections, original implementer specification. registry ↩a ↩b ↩c ↩d

[5] John P. Snyder, Map Projections: A Working Manual, USGS Professional Paper 1395 (1987), Equidistant Cylindrical section pp.90–91. Original USGS publication. The large report did not render in the present web inspection; equations were checked against the primary PROJ and PDS specifications below. registry