Lambert cylindrical equal-area projection

The Lambert cylindrical equal-area projection is a type of map projection that represents the Earth's surface on a cylinder tangent to the equator, preserving the relative areas of all regions while distorting shapes and angles, particularly increasing toward the poles.^[1] It features meridians as equally spaced vertical lines and parallels as horizontal lines spaced according to the sine of latitude to maintain equal-area properties.^[1] Developed by the Swiss mathematician Johann Heinrich Lambert in 1772 as part of his work Beiträge zum Gebrauche der Mathematik und deren Anwendungen, this projection was one of several equal-area designs he proposed, emphasizing mathematical precision for thematic mapping.^[1] Lambert's formulation used a spherical model with the standard parallel at the equator, making it a foundational cylindrical equal-area method that influenced later variants.^[2] Key properties include its equal-area (authalic) nature, ensuring no distortion in region sizes, though scale is true only along the equator and standard parallels, with shape distortion worsening at higher latitudes where the poles appear as lines equal in length to the equator.^[1] For the spherical case, forward projection equations are x = R (\lambda - \lambda_0) \cos \phi_s and y = R \sin \phi / \cos \phi_s, where R is the Earth's radius, \lambda and \phi are longitude and latitude, \lambda_0 is the central meridian, and \phi_s is the standard parallel (typically 0°).^[3] Inverse equations recover \phi = \arcsin(y \cos \phi_s / R) and \lambda = \lambda_0 + x / (R \cos \phi_s).^[3] Ellipsoidal versions adjust for Earth's oblateness using authalic latitude, involving eccentricity terms for greater accuracy.^[1] This projection is particularly suited for world-scale thematic maps, such as those showing population density, land use, or climate data, where area accuracy is paramount over shape fidelity, and for equatorial regions like Indonesia or the Pacific Ocean at large scales.^[2] Notable variants include the Behrmann projection (standard parallel at 30°) for mid-latitude balance and the Gall-Peters (45°), though the original Lambert form remains ideal for global overviews near the equator due to minimal distortion there.^[4]

Description and Properties

General Characteristics

The Lambert cylindrical equal-area projection is a type of cylindrical map projection designed to preserve the relative areas of regions on the Earth's surface. It achieves this by projecting the globe onto a cylinder that is tangent to the equator, resulting in meridians represented as equally spaced vertical straight lines and parallels as horizontal straight lines whose spacing is adjusted to maintain area integrity.^[4]^[5] In its standard form, the equator acts as the reference standard parallel, where the projection is undistorted, and the spacing of parallels increases sinusoidally with latitude to ensure equal-area preservation. This configuration produces a rectangular graticule that extends from pole to pole, with the poles mapped as horizontal lines of the same length as the equator. The overall appearance is a simple, rectilinear world map well-suited for global thematic representations.^[6]^[7] The projection's equal-area property guarantees that the sizes of landmasses and other features are depicted accurately in proportion to one another—for instance, Africa appears appropriately larger than Greenland, reflecting their true areal relationship. This characteristic makes it particularly valuable for applications requiring reliable size comparisons, such as thematic mapping of distributions like population or resources. Developed by Johann Heinrich Lambert in 1772, it remains a foundational tool in cartography for balancing area fidelity with cylindrical simplicity.^[4]^[5]^[7]

Distortions and Preservation

The Lambert cylindrical equal-area projection maintains the size of all regions on the map exactly proportional to their actual areas on the sphere, achieving this through a deliberate adjustment of scale factors along meridians and parallels such that their product equals 1 everywhere, resulting in a constant areal scale of unity.^[1] This equal-area property ensures no distortion in the representation of relative landmass sizes, even as other properties are compromised.^[8] Although areas are preserved, the projection introduces significant shape distortions, being conformal—meaning it preserves local angles and shapes—only along the equator, its standard parallel.^[1] Away from the equator, shearing effects increase progressively toward the poles, causing polar regions to appear stretched horizontally and leading to noticeable deformation of continental outlines at high latitudes.^[8] These shape distortions arise because the projection prioritizes area fidelity over angular accuracy, with maximum angular deformation quantified by formulas involving differences in principal scale factors.^[1] Distance and scale distortions also vary systematically: the projection maintains true scale along the equator and any designated standard parallels, but the meridional scale decreases poleward, while the parallel scale increases accordingly to uphold area preservation.^[1] This variation results in the elongation of high-latitude features, such as Greenland or Antarctica, appearing wider than they are tall, with scale factors becoming infinite at the poles and extreme distortions at 90° from the central meridian.^[9] Tissot's indicatrix, which visualizes local distortions by projecting infinitesimal circles from the sphere onto the map, reveals these effects specifically for the Lambert projection: such circles remain circular only at the equator, transforming into ellipses with a horizontal major axis at higher latitudes, where the eccentricity reflects the growing disparity between latitudinal and longitudinal scales while preserving the enclosed area.^[1]^[8] The consistent area of these ellipses across the map underscores the projection's equal-area integrity, even as shape and scale anisotropies intensify.^[9]

History

Invention and Development

The Lambert cylindrical equal-area projection was invented by the Swiss mathematician Johann Heinrich Lambert in 1772, as part of his pioneering efforts in developing a series of innovative map projections that included transverse and oblique variants to address limitations in earlier cylindrical designs.^[1] This projection, the first of its kind to achieve equal-area preservation on a cylindrical surface, represented a deliberate shift from the conformal focus of Gerardus Mercator's 1569 projection, prioritizing accurate representation of areal relationships over angular fidelity to better suit thematic and global mapping needs.^[1] Lambert's broader mathematical contributions informed this work by emphasizing geometric rigor in transforming spherical surfaces onto developable forms.^[1] Lambert explicitly described the cylindrical equal-area variant in his 1772 publication Beiträge zum Gebrauche der Mathematik und deren Anwendung (Part III, Section 6), where it appeared as the third of seven new projections he introduced, alongside others such as the azimuthal equal-area projection.^[1] In this treatise, published in Berlin, he detailed the projection's mechanics, adapting principles from the sinusoidal projection to ensure global area conservation through sinusoidal spacing of parallels while maintaining equally spaced meridians.^[1] This distinguished it fundamentally from equidistant cylindrical projections, like the plate carrée, which treat latitudes uniformly but introduce areal distortions at higher latitudes.^[1] Subsequent development in the 19th century built on Lambert's foundation, with Scottish mathematician James Gall refining equal-area cylindrical approaches in 1855 and further in 1885 through his equal-area cylindrical projection (later known as the Gall-Peters projection), which adjusted standard parallels (often at 45° N and S) to balance equal-area properties with reduced shape distortion for practical world mapping.^[1] These adaptations popularized variants of Lambert's design, evolving it into modern forms that retained the core sinusoidal parallel spacing for equal-area fidelity while accommodating specific regional or thematic requirements.^[1]

Notable Uses

The Lambert cylindrical equal-area projection gained prominence in the 19th century through modifications that facilitated its use in atlases for world distribution maps, particularly those emphasizing themes like climate and vegetation, where accurate area representation was essential to avoid misrepresenting regional extents. James Gall's 1855 variant, with standard parallels at 45° N and S, was employed in educational and commercial atlases to depict global patterns without distorting relative sizes of landmasses or biomes.^[1] In the 20th century, the projection became a standard in various equal-area designs and was adopted by the International Map of the World project in 1962 for equal-area global representations between 84° N and 80° S latitudes. The U.S. Geological Survey extensively utilized it during this period for thematic applications, including the 1970 National Atlas maps at scales of 1:6,000,000 and 1:10,000,000 (with standard parallels at 37° and 65° N) to illustrate U.S. resource distributions, as well as polar expedition maps at 1:39,000,000 and the 1978 Hydrocarbon Provinces map at 1:20,000,000. It also served as the basis for aeronautical charts, such as 1:1,000,000 regional world series and 1:500,000 sectional charts, and was incorporated into State Plane Coordinate Systems for east-west oriented states like North Carolina since 1933.^[1] Contemporary applications continue to leverage the projection's equal-area properties in organizations focused on global thematic mapping, including the National Geographic Society for collaborative planetary visualizations like revised Moon maps, where area preservation aids in depicting surface features accurately. It is preferred for world maps illustrating population density, land use, and environmental data by entities such as the United Nations, ensuring that size distortions do not skew interpretations of socioeconomic or ecological metrics. In GIS software, the Lambert cylindrical equal-area projection is a default option for equal-area world views in systems like ArcGIS and QGIS, supporting analyses of global datasets without area bias. Additionally, it appears in climate modeling visualizations, such as those mapping weather patterns or vegetation zones, where maintaining proportional land coverage is critical for model validation and scenario projection.^[1]^[2]^[10]

Mathematical Definition

Forward Projection Formulas

The forward projection formulas for the Lambert cylindrical equal-area projection map spherical coordinates of latitude \phi and relative longitude \lambda - \lambda_0 (where \lambda_0 is the central meridian) to Cartesian coordinates x and y on a plane tangent to a sphere of radius R, typically the authalic radius for Earth models. This normal-aspect cylindrical projection places the standard parallel at latitude \phi_s (often the equator, \phi_s = 0^\circ), with true scale along that parallel.^[1] The standard equations, assuming the origin at the equator along the central meridian, are given by

x = R (\lambda - \lambda_0) \cos \phi_s

y = R \frac{\sin \phi}{\cos \phi_s}

where \lambda and \phi are in radians.^[1] When \phi_s = 0^\circ, these simplify to x = R (\lambda - \lambda_0) and y = R \sin \phi, with the equator as both the standard parallel and the map's horizontal axis.^[11] These formulas arise from the general principles of cylindrical projections, modified to preserve area. Meridians project as equally spaced vertical lines, with spacing set to R \cos \phi_s per radian of longitude to ensure unit scale along the standard parallel. Parallels project as horizontal lines, with vertical spacing adjusted via integration to maintain equal areas: the infinitesimal area element on the sphere, R^2 \cos \phi \, d\phi \, d\lambda, must equal the planar element dx \, dy. With dx = R \cos \phi_s \, d\lambda, it follows that dy = R (\cos \phi / \cos \phi_s) \, d\phi. Integrating from the equator (\phi = 0, y = 0) yields y = (R / \cos \phi_s) \sin \phi.^[1] The scale factors confirm the equal-area property. The longitudinal scale factor is k_x = \cos \phi_s / \cos \phi, reflecting the fixed meridian spacing relative to the sphere's shrinking parallels. The latitudinal scale factor is k_y = \cos \phi / \cos \phi_s, derived from the derivative dy / (R \, d\phi). Their product is k_x k_y = 1 everywhere, ensuring no areal distortion.^[1] The poles map to horizontal lines at y = \pm R / \cos \phi_s, spanning the full width of the map (from x = -\pi R \cos \phi_s to x = \pi R \cos \phi_s). This results in severe shape distortion at high latitudes, as the pole's true point collapses into a line of length $2\pi R \cos \phi_s, though areas remain preserved.^[1]

Inverse Projection Formulas

The inverse projection for the Lambert cylindrical equal-area projection transforms Cartesian coordinates (x, y) back to geographic coordinates (latitude \phi, longitude \lambda) on the sphere of radius R, assuming a standard parallel at latitude \phi_s. The formulas are derived directly from inverting the forward projection equations, which map longitude linearly and latitude via the sine function scaled by the standard parallel. Specifically, the forward equations are x = R \cos \phi_s (\lambda - \lambda_0) and y = R \sin \phi / \cos \phi_s, where \lambda_0 is the central meridian longitude.^[1] To obtain the inverse, solve for \lambda from the x-equation: \lambda = \lambda_0 + x / (R \cos \phi_s). For latitude, rearrange the y-equation using the trigonometric identity \sin \phi = y \cos \phi_s / R, yielding \phi = \arcsin(y \cos \phi_s / R). These steps ensure the equal-area property is preserved in reverse, as the sine function maintains the integral relationship for areal conservation along parallels. When \phi_s = 0 (equator as standard parallel), the formulas simplify to \lambda = \lambda_0 + x / R and \phi = \arcsin(y / R), reducing computational complexity without loss of invertibility.^[1] Special cases arise at the projection boundaries. Longitude \lambda wraps at \pm \pi (or \pm 180^\circ) to represent the full globe, with the projected x-range spanning [- \pi R \cos \phi_s, \pi R \cos \phi_s]. Latitude is bounded at \pm \pi/2 (\pm 90^\circ); the poles are recovered exactly when |y| \geq R / \cos \phi_s, setting \phi = \operatorname{sign}(y) \cdot \pi/2 and \lambda undefined or arbitrary at the pole, though implementations often retain the input \lambda. For \phi_s \neq 0, the scaling factor \cos \phi_s adjusts the effective cylinder radius, ensuring the standard parallel remains distortion-free in both forward and inverse transformations.^[1]

Applications and Comparisons

Cartographic Applications

The Lambert cylindrical equal-area projection is particularly well-suited for thematic mapping, where preserving the relative sizes of areas is essential to avoid misleading visual interpretations of data. It is commonly employed in choropleth maps displaying global quantitative information, such as population density, economic indicators like GDP normalized by land area, or environmental metrics including deforestation rates and land cover changes, ensuring that larger regions do not appear disproportionately influential due to projection artifacts.^[12]^[2] This equal-area property facilitates accurate statistical comparisons and supports applications in climate modeling by maintaining proportional area relationships across hemispheres.^[12] In geographic information system (GIS) software, the projection benefits from widespread built-in support, enabling seamless implementation for mapping workflows. ArcGIS Pro includes the cylindrical equal-area projection with customizable parameters, such as the standard parallel (e.g., φ₀ = 45° for a Gall variant focused on mid-latitudes) and central meridian, allowing users to tailor it for specific regional emphases.^[2] Similarly, QGIS supports it through the PROJ library via EPSG code 9834 (spherical form), facilitating its use in open-source environments for vector and raster data processing.^[10] Python's Cartopy library also provides the LambertCylindrical class, which accepts parameters like central_longitude for programmatic generation of maps in scientific visualizations. In digital cartography, the projection's rectangular grid structure offers efficiency for handling raster datasets, making it advantageous for processing large-scale global imagery and overlays, such as satellite-derived land cover or vegetation indices.^[12] This grid alignment simplifies alignment with other data layers in web-based mapping applications, where equal-area preservation aids in thematic representations of environmental or socioeconomic data without requiring complex resampling.^[2] Despite these strengths, practical limitations arise in certain contexts; the projection distorts distances and shapes increasingly away from the standard parallel, rendering it unsuitable for navigation or applications requiring precise rhumb lines or great-circle routes.^[2] For polar regions, where distortion becomes extreme (with poles compressed to lines), it is often modified into interrupted variants or combined with azimuthal projections to mitigate inaccuracies in high-latitude representations.^[12]

Comparison with Other Projections

The Lambert cylindrical equal-area projection differs fundamentally from the Mercator projection in its preservation properties: while the Mercator is conformal, maintaining angles and shapes locally for navigational purposes, the Lambert preserves areas accurately but distorts shapes, particularly at higher latitudes where polar regions appear sheared rather than the extreme enlargement seen in Mercator maps like Greenland's disproportionate size.^[1] This makes the Lambert preferable for thematic world maps emphasizing distributions, such as population or climate data, over Mercator's suitability for route plotting.^[2] Compared to the Gall–Peters projection, another cylindrical equal-area variant, the Lambert uses a standard parallel at the equator for its tangent cylinder, resulting in equal spacing of parallels proportional to the sine of latitude and smoother transitions at low latitudes, whereas Gall–Peters selects standard parallels at 45° N and S to minimize mid-latitude distortion at the expense of greater shape elongation near the equator and poles.^[1] Both maintain straight, equally spaced meridians, but the Lambert's equatorial focus leads to less extreme polar compression in some applications, though Gall–Peters is often favored for its balanced global view in educational contexts.^[13] In contrast to the sinusoidal projection, which is also equal-area but pseudocylindrical with straight parallels and sinusoidally curved meridians that converge at the poles, the Lambert retains fully cylindrical geometry with straight, parallel meridians throughout, facilitating rectangular global layouts but introducing more uniform shearing distortion across latitudes rather than the sinusoidal's variable meridian spacing that reduces shape distortion near the equator.^[1] The sinusoidal thus offers better continental outlines in equatorial zones, while the Lambert excels in simplicity for graticule-based thematic mapping. Overall, the Lambert cylindrical equal-area projection trades shape fidelity—compromised by increasing angular and scale distortions toward the poles—for precise area preservation, outperforming conformal projections like Mercator in equity for global data visualization but yielding to pseudocylindrical alternatives like the sinusoidal in minimizing form distortion for continental-scale representations; it is less ideal for small-scale regional or navigational uses where azimuthal equal-area projections might better balance these properties.^[1]^[2]