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Equal-area projection

An equal-area projection, also known as an equivalent, homolographic, authalic, or equiareal projection, is a cartographic method that preserves the relative sizes of areas on the Earth's surface, ensuring that any region on the map corresponds exactly to its actual area regardless of location, though shapes, angles, and distances are typically distorted.^[1] These projections are essential for thematic mapping, such as displaying population density, land use, or resource distribution, where accurate area comparisons are critical, as opposed to conformal projections that prioritize shape preservation.^[1] Developed over centuries, equal-area projections trace their origins to ancient applications like Egyptian star charts, with significant advancements in the 18th and 19th centuries by mathematicians including Johann Heinrich Lambert, who introduced the Lambert Azimuthal Equal-Area in 1772, and Heinrich C. Albers, who devised the Albers Equal-Area Conic in 1805.^[1] Notable examples include the Lambert Cylindrical Equal-Area projection, which features straight meridians and unequally spaced parallels for global use; the Sinusoidal projection, a pseudocylindrical design dating to the mid-16th century and employed for continental maps of Africa and South America; and the Mollweide projection, introduced by Carl B. Mollweide in 1805 for world maps emphasizing elliptical meridians.^[1] While distortion increases away from central lines, standard parallels, or focal points—often resulting in stretched polar regions or elliptical shapes—these projections remain vital for equitable representation in atlases, such as the U.S. National Atlas, and continue to influence modern cartography.^[1]

Fundamentals

Definition

An equal-area projection, also known as an equivalent projection, is a type of map projection designed to preserve the relative sizes of areas on the Earth's surface, ensuring that the area of any region depicted on the map is proportional to its corresponding area on the globe, even if shapes, angles, distances, or directions are distorted.^[2] This property makes such projections particularly useful for thematic mapping where accurate representation of spatial extent is essential, without regard for angular fidelity.^[3] In contrast to conformal projections, which maintain local angles and shapes at the expense of area, or equidistant projections, which preserve distances along specified lines or directions but may alter areas, equal-area projections specifically prioritize the constancy of the areal scale factor across the map.^[4] The core geometric principle underlying this preservation involves a mapping function where an infinitesimal area element dA on the sphere corresponds to an infinitesimal area dA' on the plane such that dA' = k \cdot dA, with k being a constant areal scale factor; equivalently, in Tissot's indicatrix framework, the product of the meridional and parallel scale factors h \cdot k = 1 (or a constant) everywhere except possibly at singularities like poles.^[3]^[5] Area preservation in these projections prevents systematic misrepresentations of landmass sizes, such as the severe exaggeration of polar regions—for instance, Greenland appearing roughly the same size as Africa on the Mercator projection, despite being about one-fourteenth its area—thus providing a more balanced visual portrayal of global distributions.^[6] The systematic development of equal-area projections dates to Johann Heinrich Lambert's work in 1772.^[7]

Importance in Cartography

Equal-area projections play a crucial role in thematic cartography, where accurate representation of spatial distributions is essential for interpreting data such as population densities, land use patterns, or resource allocations. By preserving the relative sizes of areas on the map, these projections ensure that visual comparisons reflect true proportions, avoiding misleading interpretations that could arise from distorted scales in other projection types. For instance, they are particularly valuable for choropleth maps displaying variables like economic output per region or environmental indicators, as the area-based encoding aligns directly with the data's quantitative nature.^[8]^[9] One key advantage of equal-area projections is their ability to mitigate common perceptual biases introduced by non-equal-area maps, such as the Mercator projection, which disproportionately enlarges high-latitude regions and fosters misconceptions about global land distribution. A prominent example is the frequent overestimation of Greenland's size relative to Africa; in reality, Africa's land area is approximately 14 times larger than Greenland's, yet Mercator depictions make them appear comparable, skewing perceptions of continental scales and geopolitical importance. This distortion can perpetuate ethnocentric views favoring northern hemispheres, whereas equal-area projections provide a more balanced portrayal that highlights the actual vastness of equatorial and southern regions.^[6]^[10] In statistical applications, equal-area projections facilitate the creation of global grids that divide the Earth's surface into cells of uniform area, enabling reliable data aggregation and analysis across diverse scales. These grids are instrumental in fields like climate modeling, where uniform cells ensure consistent representation of phenomena such as temperature variations or precipitation patterns without bias toward polar exaggeration, and in census mapping, where they support equitable population tabulation by maintaining proportional spatial units. The Icosahedral Snyder Equal Area (ISEA) grid, for example, employs a polyhedral approach based on an icosahedron to generate hexagonal cells of equal area, offering hierarchical resolution from global to local levels and minimizing edge effects in geospatial datasets. Such systems enhance computational efficiency in simulations and promote standardized data sharing in international research.^[11]^[12]^[13] By upholding true area ratios, equal-area projections contribute to global equity in cartographic representation, particularly in atlases and educational materials that aim to convey an unbiased view of the world. They counteract historical distortions that have marginalized the visual prominence of developing regions in the Global South, fostering a more inclusive understanding of geographic realities and supporting informed discussions on topics like international aid or environmental justice. This fair depiction is especially vital in pedagogical contexts, where accurate spatial awareness helps counteract ingrained misconceptions and promotes cultural sensitivity.^[10]^[9]

Historical Development

Early Concepts

The early concepts of equal-area projections trace back to ancient cartography, where initial recognition of area distortion issues emerged without achieving full preservation. In the 2nd century AD, Claudius Ptolemy's Geographia critiqued the rectangular projection of Marinus of Tyre for producing severe area distortions, as parallels of equal length underestimated equatorial distances by one-fifth but overestimated polar distances by four-fifths, highlighting the challenges of flattening the spherical Earth.^[14] Ptolemy's own conic projections, featuring straight converging meridians and curved parallels, were not equal-area but represented a foundational effort to reduce such distortions, fostering early awareness of the need for balanced spatial representation.^[14] Renaissance cartographers in the 16th century built on this awareness, explicitly addressing area problems in cylindrical projections amid the era's voyages of discovery. Gerardus Mercator's 1569 conformal cylindrical projection preserved angles for navigation but introduced pronounced area exaggerations at higher latitudes, rendering polar regions like Greenland vastly oversized relative to equatorial areas.^[15] Abraham Ortelius, collaborating closely with Mercator, responded in his 1570 atlas Theatrum orbis terrarum by developing an oval projection with curved meridians and equidistant parallels, aiming to mitigate these imbalances and provide more proportionate global views for exploration and trade.^[15] These innovations signaled growing calls for projection methods that reconciled area accuracy with navigational utility. The 18th century marked a pivotal advancement with Johann Heinrich Lambert's 1772 introduction of the first equal-area azimuthal projection in Beiträge zum Gebrauche der Mathematik und deren Anwendungen. Developed to produce undistorted maps for scientific purposes, this projection preserved exact areas from a polar center, driven by astronomical requirements for accurate celestial charting and terrestrial demands for reliable hemispheric mapping during global expeditions.^[16] This progression reflected cartography's broader evolution from ornate, symbolic depictions to precise, mathematics-based tools, prioritizing area fidelity to support navigation, territorial analysis, and exploration in an age of empirical discovery.^[17]

Key Contributions

In the 19th century, significant advancements in equal-area projections addressed practical needs for mapping continents and world regions, building briefly on foundational concepts from Johann Heinrich Lambert's era. Heinrich Christian Albers introduced the equal-area conic projection in 1805, which preserved areas while minimizing distortion for mid-latitude regions, making it suitable for continental mapping such as North America.^[1] This projection featured two standard parallels where scale was true, and it was later revived by the U.S. Coast and Geodetic Survey in the early 20th century for official topographic work.^[1] Concurrently, Karl Brandon Mollweide developed a pseudocylindrical equal-area projection in 1805, elliptical in outline with straight parallels and curved meridians converging at the poles, ideal for thematic world maps due to its balanced distortion distribution.^[1] James Gall advanced cylindrical equal-area designs in 1855 by selecting standard parallels at 45° north and south, reducing shape distortion compared to Lambert's original while maintaining area preservation, particularly for hemispheric and global views.^[1] The 20th century saw further refinements emphasizing utility in thematic cartography and global representation. John Paul Goode's 1916 homolosine projection combined sinusoidal and Mollweide elements into an interrupted pseudocylindrical form, preserving areas exactly while interrupting the map along oceanic regions to minimize landmass distortion, widely adopted for world distribution maps.^[1] Max Eckert proposed the Eckert IV and VI pseudocylindrical equal-area projections in 1906, both elliptical with sinusoidal-like meridians, gaining popularity in American atlases for their aesthetic balance and low overall distortion in mid-latitudes.^[1] Walter Behrmann's 1910 cylindrical equal-area variant, with standard parallels at 30° north and south, optimized for equatorial emphasis in world maps, while Arno Peters' 1967 modification at approximately 45°-47° parallels sparked debates on equitable global representation despite criticisms of shape exaggeration in polar regions.^[1] John P. Snyder made pivotal contributions to equal-area projections for digital applications from 1987 to 1992, developing polyhedral frameworks to support geographic information systems (GIS). His 1992 icosahedral Snyder equal-area (ISEA) projection mapped the sphere onto an icosahedron with 20 equilateral triangular faces, ensuring exact area preservation and low distortion for hierarchical global grids, facilitating discrete global data structures in environmental modeling.^[18] This built on his earlier work, including adaptations of azimuthal equal-area methods for polyhedral globes, enabling seamless transitions between local and global scales in computer-based mapping.^[18] Post-World War II, institutional efforts standardized equal-area methods to enhance international data sharing and comparability. The U.S. Geological Survey (USGS) played a central role through Snyder's comprehensive manual (1987), which documented and recommended projections like Albers and sinusoidal for national atlases and thematic mapping, influencing global GIS standards.^[1] The International Geographical Union (IGU), via its commissions on cartography established in the 1950s, promoted consistent use of equal-area projections such as Mollweide and Goode's homolosine for international thematic maps, facilitating post-war collaboration on global resource and population data across member nations.^[19] In the 21st century, the Equal Earth projection emerged as a notable advancement, developed in 2018 by Bojan Šavrič, Bernhard Jenny, and Tom Patterson. This equal-area pseudocylindrical projection was designed to provide a visually balanced alternative to the non-equal-area Robinson projection, preserving areas while minimizing shape distortion for world maps, and has been adopted by organizations such as National Geographic for thematic and general cartography.^[20]

Mathematical Principles

Area Preservation Condition

An equal-area projection requires that the infinitesimal area element on the sphere, given by dA_{\text{sphere}} = R^2 \cos \phi \, d\phi \, d\lambda where R is the Earth's radius, \phi is latitude, and \lambda is longitude, maps to an equal area element on the plane dA_{\text{plane}} = dx \, dy. This preservation is achieved through the Jacobian determinant of the coordinate transformation from spherical coordinates (\phi, \lambda) to Cartesian plane coordinates (x, y), ensuring dA_{\text{plane}} = k \, dA_{\text{sphere}} for some constant scale factor k.^[1] In differential geometry, the transformation is defined by x = x(\phi, \lambda) and y = y(\phi, \lambda), with the Jacobian determinant J = \left| \frac{\partial(x, y)}{\partial(\phi, \lambda)} \right| = \left| \frac{\partial x}{\partial \phi} \frac{\partial y}{\partial \lambda} - \frac{\partial x}{\partial \lambda} \frac{\partial y}{\partial \phi} \right|. For area preservation on a sphere, this must satisfy J = k R^2 \cos \phi, where the cosine term accounts for the converging meridians at higher latitudes; on an ellipsoid, adjustments incorporate the flattening via authalic latitude to maintain equivalence.^[1]^[21] The derivation follows from equating the first fundamental forms of the sphere and plane metrics, leading to a partial differential equation that constrains the projection functions.^[22] This condition guarantees global and local area integrity, meaning the area of any region—such as a spherical cap or graticule polygon—remains unchanged relative to its true size, scaled uniformly by k. However, it permits local shape stretching or shearing, as the projection does not preserve angles or lengths individually; for instance, meridians and parallels may intersect at non-right angles, distorting circular graticules into ellipses of equal area but varied aspect ratios.^[1] To verify total area equality, integrate the Jacobian over a map region: \iint_{\text{[map](/page/Map)}} dx \, dy = k \iint_{\text{[sphere](/page/Sphere)}} R^2 \cos \phi \, d\phi \, d\lambda, which equals k times the spherical area (e.g., $2\pi R^2 (1 - \sin \phi_1) for a cap from pole to latitude \phi_1); choosing k = 1 or normalizing the map bounds ensures the entire projection covers the sphere's surface area $4\pi R^2 without surplus or deficit.^[1]^[22] The pursuit of such conditions traces briefly to Johann Heinrich Lambert's 1772 development of area-preserving azimuthal maps.^[1]

Projection Derivations

The general approach to deriving equal-area projections involves solving the area preservation condition through integral geometry, beginning with the sphere's differential metric ds^2 = R^2 (d\phi^2 + \cos^2 \phi \, d\lambda^2), where R is the sphere's radius, \phi is latitude, and \lambda is longitude.^[1] This metric yields the spherical area element dA = R^2 \cos \phi \, d\phi \, d\lambda, which must map to the planar area element dx \, dy such that the Jacobian determinant equals 1 (in normalized units) to preserve areas.^[1] Derivations typically integrate this element along meridians or parallels to determine coordinate functions x(\phi, \lambda) and y(\phi, \lambda), ensuring cumulative areas from the pole or equator match without distortion.^[22] For instance, the total area between latitudes \phi_1 and \phi_2 is $2\pi R^2 (\sin \phi_2 - \sin \phi_1), which guides the scaling of projected strips.^[1] In the azimuthal case, the projection centers on a pole with radial symmetry, expressed in polar coordinates as x = \rho(\phi) \cos \lambda, y = \rho(\phi) \sin \lambda, where \rho(\phi) is derived to satisfy area preservation (here using colatitude \theta = \frac{\pi}{2} - \phi for convenience, with \theta = 0 at the pole).^[1] The area element on the sphere is dA = R^2 \sin \theta \, d\theta \, d\lambda, and on the map dA' = \rho \, d\rho \, d\lambda. Setting them equal gives \rho \, d\rho = R^2 \sin \theta \, d\theta.^[22] Integrating from the pole (\theta = 0) yields \int_0^\rho \rho' \, d\rho' = \int_0^\theta R^2 \sin \theta' \, d\theta', simplifying to \frac{1}{2} \rho^2 = R^2 (1 - \cos \theta).^[1] Thus, \rho^2 = 2 R^2 (1 - \cos \theta), and using the identity $1 - \cos \theta = 2 \sin^2(\theta/2), the solution is \rho(\theta) = 2 R \sin(\theta/2). In terms of latitude \phi, \rho = 2 R \sqrt{1 - \sin \phi}. The inverse follows as \theta = 2 \arcsin(\rho / (2R)), confirming area equivalence for the full disk projection.^[22] For cylindrical projections, the derivation outlines a constant scale along meridians with variable spacing of parallels to maintain equal areas, leading to sinusoidal forms for y(\phi).^[1] Assuming equidistant meridians, x = k (\lambda - \lambda_0) where k is a scale constant, the area strip between \phi and \phi + d\phi on the sphere is $2\pi R^2 \cos \phi \, d\phi, mapping to $2\pi k \, dy on the cylinder.^[22] Setting these equal yields k \frac{dy}{d\phi} = R^2 \cos \phi, so \frac{dy}{d\phi} = \frac{R^2 \cos \phi}{k}.^[1] Integrating gives y(\phi) = \frac{R^2}{k} \sin \phi + c, with c = 0 for equator-centered projections. For the standard case with k = R, this results in y(\phi) = R \sin \phi.^[1] A standard parallel \phi_s adjusts k = R \cos \phi_s for regional balance, yielding y(\phi) = R \frac{\sin \phi}{\cos \phi_s}.^[22] The authalic latitude \eta simplifies equal-area computations, especially in digital projections, by transforming geodetic latitude \phi to an equivalent spherical latitude preserving meridional areas.^[1] For a sphere (e = 0), it follows from integrating the parallel circumference factor: \eta = \arcsin \left( \int_0^\phi \cos \phi' \, d\phi' \right) = \arcsin(\sin \phi) = \phi.^[22] This substitution linearizes the area integral in cylindrical or other forms, enabling efficient numerical implementation by treating the surface as a unit sphere.^[1] For ellipsoids, extensions incorporate eccentricity via q(\phi) = \int_0^\phi (1 - e^2) / (1 - e^2 \sin^2 \phi')^{3/2} \, d\phi', with \eta = \arcsin(q / q_p) where q_p is the polar value, but spherical cases prioritize the cosine integral for baseline derivations.^[1]

Types of Equal-area Projections

Cylindrical and Pseudocylindrical

Cylindrical equal-area projections map the spherical surface onto a cylinder tangent to the equator or at specified standard parallels, unrolling it into a rectangular grid while preserving areas. In the standard form, known as the Lambert cylindrical equal-area projection, meridians project as equally spaced vertical lines, and parallels as horizontal lines spaced according to the sine of latitude to ensure area preservation. The forward transformation equations for a sphere of radius R, with central meridian \lambda_0 and standard parallel \phi_s (often 0° for the equatorial case), are given by

x = R (\lambda - \lambda_0) \cos \phi_s, \quad y = R \frac{\sin \phi}{\cos \phi_s},

where \lambda and \phi are longitude and latitude in radians.^[1] This formulation maintains true scale along the standard parallel(s) and results in significant shape distortions, particularly elongation of features near the poles, where vertical stretching becomes extreme.^[1] A notable variant is the Gall–Peters projection, developed by James Gall in 1855 and later promoted by Arno Peters in 1973. It sets the standard parallels at 45° N and S, yielding

x = R (\lambda - \lambda_0) \cos 45^\circ = R (\lambda - \lambda_0) \frac{\sqrt{2}}{2}, \quad y = R \frac{\sin \phi}{\cos 45^\circ} = R \sqrt{2} \sin \phi.

This adjustment aims to balance distortions between low and high latitudes but introduces severe east-west compression at the equator, making equatorial regions appear unnaturally tall and narrow, a criticism often leveled against its use in world maps.^[23] Despite these issues, it underscores the trade-offs in cylindrical designs, where area fidelity prioritizes over shape integrity.^[1] Pseudocylindrical equal-area projections relax the strict cylindrical geometry by allowing curved meridians, typically sinusoidal in form, while keeping parallels straight and horizontal; this produces a more compact global representation within an oval or rectangular boundary. The sinusoidal projection exemplifies this class, with meridians as sine curves except for the straight central meridian, and equations

x = R (\lambda - \lambda_0) \cos \phi, \quad y = R \phi.

Its area preservation arises from the cosine scaling in the x-coordinate matching the diminishing parallel lengths on the sphere, ensuring true scale along the central meridian and all parallels.^[1] Originating in the 16th century and refined over time, it avoids the infinite extension of pure cylindrical maps at the poles.^[1] Another key pseudocylindrical example is the Mollweide projection, introduced by Karl B. Mollweide in 1805. It projects the globe into an ellipse with a 2:1 axial ratio, where meridians are equally spaced elliptical arcs and parallels are straight lines. The forward equations involve solving the transcendental relation $2\theta + \sin 2\theta = \pi \sin \phi for \theta, followed by

x = \frac{2 \sqrt{2} R}{\pi} (\lambda - \lambda_0) \cos \theta, \quad y = \sqrt{2} R \sin \theta,

which requires numerical iteration but guarantees equal areas through the integral properties embedded in the \theta definition.^[1] Though not directly using elliptic integrals in its primary formulation, the projection's design invokes similar mathematical constructs for maintaining equivalence.^[1] These cylindrical and pseudocylindrical equal-area projections are particularly suited for world-scale thematic mapping, such as choropleth displays of population or resource distribution, due to their rectangular or near-rectangular formats that facilitate graticule overlays.^[1] However, distortions intensify toward the poles, with angular deformations exceeding 100% in high latitudes, limiting their utility for navigation or precise distance measurements; they remain staples in atlases for global thematic purposes where area accuracy is paramount.^[1]

Conic and Azimuthal

Conic equal-area projections map the spherical surface onto a secant cone, making them suitable for regions with east-west extents at mid-latitudes, where distortion is minimized between two standard parallels. The Albers equal-area conic projection, developed by Heinrich C. Albers in 1805, serves as a standard for such maps, particularly for the conterminous United States.^[1] In this projection, parallels are arcs of concentric circles unequally spaced from the apex, while meridians radiate at equal intervals from the apex.^[1] The forward formulas for a sphere are given by:

\rho = \frac{R \sqrt{C - 2n \sin \phi}}{n}, \quad \theta = n (\lambda - \lambda_0),

x = \rho \sin \theta, \quad y = \rho_0 - \rho \cos \theta,

where C = \cos^2 \phi_1 + 2n \sin \phi_1, n = \frac{\sin \phi_1 + \sin \phi_2}{2}, \rho_0 is the value of \rho at the origin latitude \phi_0, R is the sphere's radius, \phi_1 and \phi_2 are the standard parallels, \phi is latitude, and \lambda - \lambda_0 is the difference in longitude from the central meridian.^[1] Scale is true along these two standard parallels (typically 29°30'N and 45°30'N for U.S. maps), and area is preserved throughout, with minimal linear distortion in the zone between them.^[1] This makes the Albers projection ideal for thematic mapping of continental-scale areas, such as national atlases or state boundaries, where accurate area representation outweighs shape fidelity.^[1] Azimuthal equal-area projections project onto a secant plane tangent at a central point, often a pole, preserving area and directions from the center, which suits polar or hemispheric views. The Lambert azimuthal equal-area projection, introduced by Johann Heinrich Lambert in 1772, is polar-focused and features a circular boundary encompassing a hemisphere.^[1] For the polar aspect on a sphere, the forward formulas are:

\rho = 2R \sin\left(\frac{\pi}{4} - \frac{\phi}{2}\right), \quad x = \rho \sin(\lambda - \lambda_0), \quad y = -\rho \cos(\lambda - \lambda_0),

or equivalently, x = 2R \sin(\phi/2) \cos \lambda, y = 2R \sin(\phi/2) \sin \lambda when centered at the pole with adjusted coordinates.^[1] Meridians appear as straight lines from the center, and parallels as concentric circles, with scale true along the central meridian in non-polar aspects but decreasing radially from the center overall.^[1] This projection excels for maps of hemispheres, continents, or polar regions, such as Arctic insets or Pacific Ocean overviews, due to its low distortion near the center and equal-area preservation for global resource depictions.^[1] Other variants include the Bonne projection, a pseudoconic equal-area type from the 18th century by Rigobert Bonne, which uses sinusoidal meridians and a single standard parallel.^[1] It maintains true scale along the central meridian and all parallels, with curved meridians converging at the poles, making it suitable for historical continental maps like those of Europe or Africa.^[1] The Hammer-Aitoff projection, developed by Ernst von Hammer in 1892 as a modification of the Aitoff projection, is an azimuthal equal-area compromise for world maps.^[1] It employs modified integrals to ensure area preservation, resulting in elliptical meridians and equally spaced parallels, though with increased shape distortion at the edges compared to pure azimuthal forms.^[1] Both conic and azimuthal equal-area projections reduce shape distortion relative to cylindrical types for their targeted zones—conic for mid-latitude continents and azimuthal for polar hemispheres—while prioritizing area accuracy over equatorial global strips.^[1]

Applications

Thematic and Statistical Mapping

Equal-area projections are essential in thematic mapping, particularly for choropleth maps that visualize variables such as population density or land use, as they maintain proportional areas to ensure that color gradients accurately represent data intensity without distortion from unequal regional sizing.^[24] In these applications, the preservation of area allows for fair comparisons across geographic units, where darker shades indicate higher densities without inflating or shrinking the visual impact of larger versus smaller regions; for instance, equal-area hexbin overlays can aggregate point data into hexagonal bins of uniform size, facilitating the display of clustered phenomena like urban growth patterns.^[25] This approach is particularly valuable for density mapping, where accurate area representation prevents misinterpretation of spatial distributions.^[24] In statistical analysis, equal-area projections support the creation of discrete global grids that enable precise data interpolation and aggregation, such as the Icosahedral Snyder Equal Area (ISEA) grid system, which tessellates the Earth's surface into hexagonal cells of equal area using an icosahedral base.^[11] The ISEA grid can be refined to higher resolutions, providing uniform sampling in climate data interpolation where variables like temperature or precipitation are averaged without bias from varying cell sizes.^[11]^[26] This tessellation minimizes edge effects and supports machine learning applications in environmental modeling by providing a consistent spatial framework for overlaying multivariate datasets.^[26] Integration of equal-area projections in geographic information systems (GIS) further enhances their utility for statistical mapping, as seen in software like ArcGIS, where tools for raster resampling project global datasets into equal-area coordinate spaces to avoid aggregation errors during analysis.^[27] In ArcGIS, the Project Raster tool ensures that output cells maintain equal area in the projected space, preserving the integrity of continuous data like elevation or vegetation indices when resampled from geographic to projected systems, thus preventing over- or under-representation in global-scale computations.^[27] This capability is critical for handling large raster datasets, where bilinear or cubic convolution resampling methods combined with equal-area projections yield reliable summaries for thematic overlays.^[28] A notable case study is the World Population Cartogram, which employs an equal-area projection to resize countries proportional to their population, transforming the map into a visual representation where land area reflects human distribution rather than physical geography, thereby highlighting disparities such as the dominance of densely populated urban centers in Asia over sparsely inhabited rural areas elsewhere.^[29] Developed using GIS tools like those in ArcGIS, this cartogram stretches nations like India and Nigeria to emphasize their demographic weight, aiding in the analysis of global urbanization trends and resource allocation needs without the biases of traditional projections.^[30] Such visualizations underscore urban-rural divides by amplifying the spatial extent of high-density regions, promoting informed discussions on equitable development.^[29]

Global Resource Distribution

Equal-area projections are essential in mapping the global distribution of natural resources such as minerals, forests, and water bodies, as they preserve the actual surface areas to enable accurate estimates of resource yields and extents. For instance, in mineral resource mapping, the interrupted Goode homolosine projection has been used to represent mining areas on a 50 × 50 km grid, ensuring undistorted area measurements for environmental impact assessments worldwide, with concentrations accurately depicted in regions like northern Chile and northeastern Australia. Similarly, global forest cover datasets employ the interrupted Goode homolosine projection to maintain proportional land coverage, facilitating reliable assessments of forest resource distribution without exaggeration of continental sizes. In water resource mapping, the Albers equal-area conic projection supports spatial data analysis for hydrological features, preserving basin and watershed areas for precise yield calculations in the United States and beyond. In climate and ecological applications, equal-area projections underpin maps of the Köppen-Geiger climate classification, where area preservation is crucial for delineating zones without skewing regional extents. The Albers equal-area conic projection, for example, is applied in high-resolution Köppen maps at 5 arc-minute scales, accurately representing climate distributions across diverse geographies like the United States to avoid misrepresenting the proportions of temperate versus arid zones. For biodiversity hotspots, the Behrmann equal-area projection grids species ranges at approximately 110 km resolution, linking geographic distributions of amphibians, reptiles, mammals, and birds to climate variables while preventing overestimation of polar biome areas, where extreme conditions already limit species richness compared to tropical regions. Satellite-derived environmental data, such as MODIS imagery, is routinely reprojected to equal-area frameworks to compute global vegetation indices like NDVI and EVI, ensuring unbiased aggregation across latitudes. NASA's MODIS vegetation index products use the sinusoidal projection for initial tiling but support reprojection to equal-area systems like the Lambert azimuthal equal-area or Goode interrupted homolosine, which handle global datasets by maintaining area integrity during mosaicking and analysis of land features. This reprojection process accommodates spherical harmonics in underlying global models to represent Earth's curvature accurately, enabling consistent temporal comparisons of vegetation conditions without polar inflation. Equal-area projections feature prominently in United Nations reports on land cover change, particularly for tracking deforestation rates free from area bias. The FAO's Global Forest Resources Assessment 2020, for instance, reprojected forest cover maps using Lambert azimuthal equal-area projection for calibration in continental analyses, supporting estimates of 420 million hectares deforested globally since 1990 (as of 2020) while preserving true land proportions for policy-relevant monitoring.^[31]^[32]

Limitations and Comparisons

Shape and Distance Distortions

In equal-area projections, shape distortion arises because the preservation of area necessitates variations in linear scale factors along different directions, leading to the deformation of circular features into ellipses as visualized by Tissot's indicatrix. Tissot's indicatrix consists of infinitesimal circles on the globe that map to ellipses on the projection plane, maintaining constant area but exhibiting unequal major and minor axes that reflect angular deformation and shearing. The eccentricity of these ellipses increases away from standard lines or points, resulting in pronounced shape alterations, such as the elongation of meridians relative to parallels or vice versa.^[1]^[3] This distortion manifests visually in specific projections, where continental outlines appear stretched or compressed; for instance, in the Gall-Peters projection, high-latitude landmasses like Greenland and Africa take on a sausage-like, vertically elongated form due to severe north-south stretching and east-west compression, with angular deformation indices reaching extremes near the poles. In azimuthal equal-area projections, such as the Lambert variant, shapes near the periphery become radially stretched, crowding and distorting features at the map's edges, with increasing distortion and tangential compression affecting equatorial and opposite polar regions. These effects stem from the projection's geometry, where scale factors along meridians (h) and parallels (k) diverge, with the maximum angular deformation ω approximated by sin(ω/2) = |h - k| / (h + k), highlighting the trade-off inherent to area preservation.^[1]^[3] Distance distortions in equal-area projections occur because geodesic distances are not preserved globally, with scale varying directionally and locationally, true only along designated standard parallels or central points. In conic equal-area projections like Albers, distances are accurate along the two standard parallels (e.g., 29.5° N and 45.5° N for continental U.S. maps), but deviate elsewhere, with meridional scale h exceeding parallel scale k or vice versa, leading to up to 1-2% variation in mid-latitudes. Cylindrical types, such as the sinusoidal projection, maintain true scale along the central meridian and parallels but introduce errors in oblique directions, amplifying distortions toward the poles where poles collapse to lines.^[1] To mitigate these distortions, cartographers select standard lines to minimize local scale variations and angular errors, often limiting the map extent to one hemisphere or using interrupted forms to reduce peripheral stretching; however, global applications inevitably involve trade-offs, as Gauss's Theorema Egregium demonstrates the impossibility of simultaneously preserving area, angles, and distances on a developable surface without distortion. For example, in the Mollweide projection, standard parallels at approximately 40.44° N/S optimize scale balance, reducing north-south stretching at the equator to about 23% while compressing polar shapes. Despite these strategies, equal-area projections prioritize areal fidelity, accepting shape and distance inaccuracies for applications like thematic mapping.^[1]^[3]

Versus Conformal Projections

Equal-area projections prioritize the preservation of accurate relative sizes of regions on the Earth's surface, ensuring that areas are represented in their true proportions regardless of location. In contrast, conformal projections, such as the Mercator projection, maintain local shapes and angles, making small features appear undistorted but leading to significant area exaggerations, particularly near the poles where landmasses like Greenland appear vastly larger than they are. This fundamental difference stems from the mathematical constraints of map projections: as established in cartographic theory, no projection can simultaneously preserve both areas and angles everywhere on the globe.^[33] The trade-offs between these projection types are pronounced in their applications. Equal-area projections are ideal for quantitative analysis, such as thematic mapping of population density or resource distribution, where preserving area ratios allows for reliable comparisons and statistical accuracy without misleading scale variations. Conformal projections, however, excel in scenarios requiring precise local geometry, like nautical navigation on the Mercator or topographic surveys using the Lambert Conformal Conic, where true directions and shapes facilitate practical measurements. Neither type fully avoids distortions—equal-area maps often elongate or compress shapes, while conformal maps sacrifice size fidelity—but the choice depends on whether the map's purpose emphasizes areal integrity or angular precision.^[33]^[34] Hybrid projections like the Robinson projection illustrate compromises beyond strict equal-area or conformal categories, neither preserving areas nor angles perfectly but aiming for a visually balanced representation of the world suitable for general reference maps. Developed in 1963, it minimizes overall distortion to provide an aesthetically pleasing global overview, though it is unsuitable for precise quantitative work. Selection criteria for projections thus hinge on the map's intent: equal-area for "what if" analyses like resource allocation modeling, where size accuracy informs decision-making, versus conformal for qualitative overviews or navigation where shape preservation aids interpretation.^[35]^[33]

References

[1]
None
Below is a merged summary of equal-area projections based on the provided segments from the USGS Professional Paper 1395. To retain all information in a dense and organized manner, I will use a combination of narrative text and a table in CSV format for key details. The narrative provides an overview, while the table captures specific details such as definitions, history, characteristics, examples, and URLs systematically.
[2]
All equal-area map projections are created equal, but some are ...
Although equal-area projections are designed to preserve area, results show variance in one-degree area after projection, and total areas vary with projection ...
[3]
[PDF] MAP PROJECTION PROPERTIES - UNC Charlotte Pages
An azimuthal projection can be equidistant, conformal or equal-area. To obtain an equal-area projection, one must preserve elementary surfaces. A surface on.
[4]
Map Projections Classification - MDPI
May 29, 2022 · Therefore, in equal-area projections, the local area scale factor is equal to one in the entire map field. In equidistant projections, the ...
[5]
[PDF] Floating Ring: A New Tool for Visualizing Distortion in Map Projections
An equivalent or equal-area projection preserves unity of the products of the scale factors at every point and in every direction on a map, except at the poles.<|control11|><|separator|>
[6]
[PDF] Projections and Coordinate Systems
Area. Increasingly distorted toward the polar regions. For example, in the Mercator projection, although. Greenland is only one-eighth the size of South ...
[7]
Lambert Azimuthal Equal-Area Projection Definition | GIS Dictionary
A planar equal-area projection usually restricted to a hemisphere, with the polar and equatorial aspects used most often. Invented by Swiss polymath Johann ...
[8]
Map Projections | Axis Maps
Projections which preserve areas are called equivalent or equal-area projections. A map projection either preserves areas everywhere, or distorts it everywhere.Projection Properties · Choosing A Projection · Projection ParametersMissing: populations | Show results with:populations
[9]
[PDF] Understanding Map Projections
Equal-area maps are invaluable for understanding the real spatial extent of regions, especially in education and thematic mapping. Equidistant Projections.
[10]
Mercator Projection
... area of Africa is 14 times greater. Another way of showing the distortion inherent in this projection is to use a watchglass with the arctic circle drawn on it.Missing: polar regions
[11]
Jack Swab, Derek Alderman in 'The Conversation': World maps get ...
Aug 28, 2025 · The Mercator projection shows Greenland as nearly the same size as Africa, when in reality Africa is about 14 times larger.
[12]
[PDF] ISEA Discrete Global Grids - GMU
The acronym ISEA stands for icosahedral Snyder equal area. The grid cells not only have equal areas, they are hexagons when projected onto an icosahedron!
[13]
Spatial analysis and visualization of global data on multi-resolution ...
The icosahedral version of the Snyder equal area polyhedral projection (Snyder 1992) is used to inversely project the cells from the icosahedral faces to the ...Missing: census | Show results with:census
[14]
[PDF] Statistical Grids for the U.S. Census Bureau
Jul 17, 2024 · Equal area grid cells. • Ability to nest hierarchically. (square/quadrilateral shape). • Easy data integration (Earth. Observation, other ...
[15]
[PDF] 11 · The Culmination of Greek Cartography in Ptolemy
89 Finally, Ptolemy's second projection, when used for world cartography, could have caused much distortion in the areas of the oikoumene farthest to the ...
[16]
[PDF] Map Projections in the Renaissance - The University of Chicago Press
AZIMUTHAL EQUIDISTANT PROJECTION. CENTERED ON THE NORTH AND SOUTH POLES BY. GIOVANNI VESPUCCI, 1524. The arrangement of the hemi- spheres in this projection is ...
[17]
[PDF] Johann Heinrich Lambert - University of St Andrews
Aug 16, 2021 · In 1762 he published seven projection methods, among others the conical projection. (preserving angle), the azimuthal projection (preserving ...<|control11|><|separator|>
[18]
Map - Cartography, Geography, Navigation | Britannica
No readable text found in the HTML.<|control11|><|separator|>
[19]
An Equal-Area Map Projection For Polyhedral Globes | Cartographica
The Lambert Azimuthal Equal-Area projection can be modified to provide an exactly fitting, perfectly equal-area projection for any polyhedral globe that has ...Missing: ISEA | Show results with:ISEA
[20]
The International Geographical Union in the Post-war Period
This chapter introduces into the history of the International Geographical Union after the Second World War and the development in the 1950s and 60s.Missing: projections | Show results with:projections
[21]
[PDF] General Theory of Equivalent Projections - National Geodetic Survey
Jan 1, 2006 · Since, when y is equal to 1, this map area must be equal to one-fourth of the area of the given sphere or rR2, we must have the equation. 4.
[22]
[PDF] Map Projection Equations - DTIC
Jun 1, 2025 · miapping schemeis. The Important unifying principles of differential geomnetry are 'applied to produce the equal area, conformal and ...
[23]
[PDF] US ~eological Survey Profes-sional-Paper 1453
This paper is an album of map projections, presenting 90 basic projections in over 130 modifications, and how they transform the sphere onto a flat plane.
[24]
Mapping the Global South: Equal-Area Projections for Choropleth ...
Our study takes the first substantial step in investigating projection choices for world choropleth maps in applied visualization research.
[25]
6 Equal Area Projections for Thematic Mapping That Reveal Hidden ...
Apr 21, 2025 · Discover 6 essential equal area projections that ensure accurate thematic mapping for population, economic, and environmental data ...
[26]
Eco-ISEA3H, a machine learning ready spatial database for ...
We present the Eco-ISEA3H database, a compilation of global spatial data characterizing climate, geology, land cover, physical and human geography.
[27]
Project Raster (Data Management)—ArcGIS Pro | Documentation
The cells of the raster dataset will be square and of equal area in map coordinate space, although the shape and area a cell represents on the surface of the ...Raster Toolset · Project Raster (data... · Parameters
[28]
Fundamentals of georeferencing a raster dataset—ArcMap
The three most common resampling techniques are nearest neighbor assignment, bilinear interpolation, and cubic convolution. These techniques assign a value to ...Fundamentals Of... · Transforming The Raster · Resampling The Raster...
[29]
The map we need if we want to think about how global living ...
Sep 12, 2018 · By showing us where the people in the world are, cartograms help us understand global living conditions better.
[30]
Remapping the World's Population - Esri
Cartograms are often used for displaying population data. The Cartogram Geoprocessing tool was developed by Esri staff member Tom Gross.
[31]
[PDF] Map Projections - USGS Publications Warehouse
A map projection is equal area if every part, as well as the whole, has the same area as the cor- responding part on the Earth, at the same ...
[32]
Selecting a Map Projection - National Geographic Education
Jan 22, 2024 · For example, conformal maps show true shapes of small areas but distort size. Equal area maps distort shape and direction but display the true ...Missing: fair | Show results with:fair
[33]
The Robinson Projection - UW-Madison Geography
The Robinson projection can best be described as being pseudocylindrical, but given its unique method of development, it does not fall perfectly into any known ...Missing: 1963 | Show results with:1963