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Understanding Equal-Area Projections in Cartography
In cartography, an equivalent, authalic, or equal-area projection is a map projection that preserves relative area measure between any and all map regions. This fundamental property makes equal-area projections essential tools for geographers, cartographers, and anyone working with spatial data where accurate size representation matters more than preserving shapes or angles.
The challenge of representing Earth’s three-dimensional spherical surface on a two-dimensional plane has occupied cartographers for centuries. Every map projection involves compromises, and understanding these trade-offs is crucial for selecting the appropriate projection for any given purpose. A map projection either preserves areas everywhere, or distorts it everywhere. This is an all-or-nothing property.
Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal. This mathematical reality means cartographers must choose which properties to preserve based on the map’s intended use. Equal-area projections sacrifice shape accuracy to maintain proportional area representation, making them invaluable for thematic mapping and statistical visualization.
The Mathematical Foundation of Equal-Area Projections
Equal-area projections work by ensuring that any given area on the map corresponds proportionally to the same area on Earth’s surface. When you examine an equal-area map, a square inch in one location represents the same amount of Earth’s surface as a square inch anywhere else on the map, regardless of latitude or longitude.
In an equal area projection, Tissot circles are all the same relative size across the map. The Tissot indicatrix is a mathematical tool cartographers use to visualize distortion in map projections. Despite how the Tissot indicatrix changes from a circle to an ellipse, this projection retains relative size. While the circles may become ellipses showing shape distortion, their areas remain constant, confirming the equal-area property.
Angles, areas, directions, shapes, and distances can become distorted when transformed from a curved surface to a plane. Different projections have been designed where the distortion in one property is minimized, while other properties become more distorted. Equal-area projections minimize area distortion at the expense of other properties, particularly shape and angular relationships.
Major Types of Equal-Area Projections
Cartographers have developed numerous equal-area projections over the centuries, each with distinct characteristics suited to different mapping needs. Understanding the major types helps in selecting the most appropriate projection for specific applications.
Cylindrical Equal-Area Projections
The most common projection surfaces are cylindrical (e.g., Mercator), conic (e.g., Albers), and planar (e.g., stereographic). Cylindrical equal-area projections imagine wrapping a cylinder around the globe, projecting the Earth’s surface onto it, then unrolling it to create a flat map.
Gall-Peters Projection: The Gall–Peters projection was first described in 1855 by the Scottish clergyman James Gall, who presented it along with two other projections at the Glasgow meeting of the British Association for the Advancement of Science (the BA). It achieved more widespread attention after Arno Peters reintroduced it in 1973.
The standard parallels of the Gall–Peters are 45° N and 45° S. This projection became the center of significant controversy in the cartographic world. He promoted it as a superior alternative to the commonly used Mercator projection, on the basis that the Mercator projection greatly distorts the relative sizes of regions on a map.
The controversy surrounding the Gall-Peters projection extended beyond technical cartography into social and political realms. When Arno Peters unveiled his projection in 1973, he claimed it was the first map to show the world fairly, accusing the Mercator of systematic bias against developing nations. Professional cartographers were outraged, pointing out that identical equal-area projections had existed since James Gall published the same mathematics in 1855.
The U.S. state of Massachusetts and Boston Public Schools began phasing in these maps in March 2017, becoming the first public school district and state in the United States to adopt Gall–Peters maps as their standard. This decision reignited debates about the role of map projections in education and social justice.
Hobo-Dyer Projection: Conceived by ODT, Inc., this map projection draws inspiration from the Peters equal area concept but is designed to minimize distortion of both size and shape of land masses, providing a more accurate representation of the relative sizes of countries and continents compared to traditional map projections. In a stark deviation, the Hobo-Dyer projection assumes a cylinder that wraps around the globe, intersecting it at 37½° north and south.
The Hobo-Dyer map projection garnered global attention when President Jimmy Carter utilized it to illustrate his agency’s international efforts during his Nobel Peace Prize ceremony in December 2002.
Conic Equal-Area Projections
Conic projections imagine placing a cone over the globe, with the cone’s surface touching the Earth along one or two standard parallels. These projections work particularly well for mid-latitude regions with greater east-west than north-south extent.
Albers Equal-Area Conic Projection: The USGS commonly uses the Albers Equal Area Conic projection because of how it proportionally represents areas for the conterminous United States. H. C. Albers first introduced this map projection in 1805 with two standard parallels (secant).
For example, distances and scale are true only on both standard parallels. Although the direction is reasonably accurate, it’s not conformal, perspective, or equidistant. The Albers projection excels at representing countries or regions that extend primarily in an east-west direction.
Some, like the Albers Equal Area Conic, distort more as you go north or south, but don’t distort much as you go east or west. So, they’re good for mapping an area like the United States. This makes it the projection of choice for many U.S. government mapping agencies.
Pseudocylindrical Equal-Area Projections
Pseudocylindrical projections represent parallels as straight lines but curve the meridians, creating a more visually appealing representation of the globe while maintaining equal-area properties.
Sinusoidal Projection: The sinusoidal projection is one of the oldest and simplest equal-area projections. To contrast, equal-area projections such as the Sinusoidal projection and the Gall–Peters projection show the correct sizes of countries relative to each other, but distort angles. In actuality, some of the oldest projections are equal-area (such as the sinusoidal projection), and hundreds more have been described.
Mollweide Projection: The Mollweide projection, also known as the Babinet or homolographic projection, presents the world in an ellipse. It maintains equal-area properties while providing a more aesthetically pleasing representation than some cylindrical equal-area projections. The projection is particularly useful for world maps showing global distributions.
Goode’s Homolosine Projection: This innovative projection combines the sinusoidal projection for low latitudes with the Mollweide projection for high latitudes. By interrupting the map along oceanic regions, Goode’s Homolosine minimizes distortion of continental landmasses while maintaining equal-area properties. This makes it especially valuable for displaying global data related to land-based phenomena.
Equal Earth Projection: The Equal Earth map projection is a new equal-area pseudocylindrical projection for world maps jointly developed by Bojan Šavrič (Esri), Tom Patterson (US National Park Service), and Bernhard Jenny (Monash University). The Equal Earth map projection was created as equal-area projection that shows land features at their true relative sizes and has been widely adopted since it was introduced in August 2018.
The Equal Earth projection was created in response to a wave of news stories in 2017 following the Boston Public Schools announcement that it was switching to the Gall-Peters projection for all classroom world maps. These articles erroneously asserted that the Gall-Peters projection was the only equal-area projection that shows land features at their true relative sizes, despite the consensus about inappropriateness of this projection for small-scale mapping.
In addition to being rigorously equal-area throughout, other Equal Earth projection features include: • An overall shape similar to that of the Robinson projection. (The Robinson, although popular and pleasing to the eye, is not equal-area as is the Equal Earth projection). This combination of mathematical rigor and visual appeal has made the Equal Earth projection increasingly popular for educational and reference mapping.
Applications and Uses of Equal-Area Projections
Equal-area projections serve critical roles in various fields where accurate representation of area is paramount. Understanding when and why to use these projections helps ensure that maps communicate information effectively and accurately.
Thematic Mapping and Statistical Visualization
Equivalent projections are widely used for thematic maps showing scenario distribution such as population, farmland distribution, forested areas, and other phenomena where area matters. When creating choropleth maps that use color to represent data density or intensity, equal-area projections ensure that visual comparisons remain valid.
If you are making choropleth or dot density maps, look for an equal-area projection. This recommendation stems from the fundamental principle that these map types rely on area to convey information. Area — Maps of density demand equal area projections. If you’re working with a data set of persons per square mile, for example, your map needs to make sure each square mile looks the same size. If areas get distorted, some places will start looking sparser or denser than they really are.
The projection is appropriate for small-scale mapping, especially for thematic world maps illustrating area characteristics and analysis requiring accurate areas. This makes equal-area projections indispensable for global analyses of climate patterns, resource distribution, agricultural land use, and demographic trends.
Geographic Information Systems (GIS)
Modern GIS applications frequently require equal-area projections for spatial analysis. When calculating areas, analyzing spatial patterns, or performing overlay operations where area measurements matter, equal-area projections provide the mathematical foundation for accurate results. Many GIS software packages include numerous equal-area projection options to accommodate different geographic extents and analytical requirements.
For regional analyses, selecting an equal-area projection appropriate to the study area’s extent and orientation is crucial. Different projections have different distortion patterns. Some, like the Albers Equal Area Conic, distort more as you go north or south, but don’t distort much as you go east or west. Understanding these distortion patterns helps GIS professionals choose projections that minimize distortion in their specific study areas.
Educational and Reference Mapping
Equal-area projections play an increasingly important role in educational settings, where accurately representing the relative sizes of countries and continents helps students develop accurate mental maps of the world. This makes it particularly useful for educational purposes, social justice initiatives, and for gaining a more accurate understanding of global geography.
The first known thematic map published using the Equal Earth projection is a map of the global mean temperature anomaly for July 2018, produced by the NASA’s Goddard Institute for Space Studies. This demonstrates how equal-area projections serve scientific communication by ensuring that spatial patterns are represented without area distortion.
Despite the academic controversy, the Peters projection was adopted by numerous international organizations. UNESCO, UNICEF, and many NGOs chose it for their publications because it shows developing nations at their true proportional size, supporting their mission to highlight global equity issues.
Advantages of Equal-Area Projections
Equal-area projections offer several compelling advantages that make them essential tools in cartography and spatial analysis. Understanding these benefits helps explain why cartographers continue to develop new equal-area projections and refine existing ones.
Accurate Size Comparisons
The primary advantage of equal-area projections is their ability to represent the relative sizes of geographic features accurately. The term “equal area” in the Hobo-Dyer Map refers to its cartographic projection, which ensures that each region’s size on the map accurately reflects its true relative size on the Earth’s surface. This property enables direct visual comparison of areas across the entire map.
This means that areas that are typically distorted, such as Greenland appearing larger than Africa on traditional maps, are accurately represented in terms of their actual land area. On a Mercator projection, Greenland appears similar in size to Africa, when in reality Africa is approximately 14 times larger. Equal-area projections correct this misrepresentation.
Statistical Integrity
For maps displaying statistical data, equal-area projections maintain the mathematical relationships between areas and the data they represent. When showing population density, resource distribution, or any other area-based statistic, these projections ensure that the visual representation accurately reflects the underlying data relationships.
This statistical integrity is particularly important for scientific research, policy analysis, and educational materials where accurate spatial understanding influences decision-making and comprehension. Maps that distort area can lead to misunderstandings about the magnitude and distribution of phenomena.
Social and Political Equity
The projection gained support among groups advocating for social justice. It also gained support for a more balanced representation of the world. By accurately portraying the sizes of developing countries, it challenges historical biases and promotes a more egalitarian perspective.
The Hobo-Dyer projection is notable for its social justice aspect. By showing all countries at their true relative sizes, equal-area projections counter the implicit hierarchies that can result from projections that exaggerate the size of wealthy northern hemisphere nations while minimizing tropical and southern hemisphere regions.
The Correct the Map campaign, backed by the African Union and the Caribbean Community, has called for the wider adoption of the Equal Earth projection as an alternative to the Mercator projection. This reflects growing recognition that map projections carry social and political implications beyond their technical properties.
Limitations and Distortions in Equal-Area Projections
While equal-area projections excel at preserving area relationships, they necessarily introduce other forms of distortion. Understanding these limitations is essential for using these projections appropriately and interpreting maps created with them.
Shape Distortion
However, representing area ratios correctly necessarily distorts shapes more than many maps that are not equal-area. This shape distortion is the most visible limitation of equal-area projections and often the source of criticism.
Shapes, directions, angles, and distances are distorted and stretched north-south in tropical and mid-latitude areas. Nearer the poles, features are compressed in the north-south direction. This distortion pattern varies depending on the specific equal-area projection used, but all equal-area projections distort shapes to some degree.
Shape Distortion: A significant drawback of the Gall-Peters projection is the distortion of shapes. While areas are accurately represented, the shapes of continents and countries are elongated and stretched. Arthur Robinson famously said that it looks like long underwear hung out on a clothesline. This vivid description captures the visual impact of extreme shape distortion in some cylindrical equal-area projections.
While the Hobo-Dyer projection minimizes distortion in terms of area, it does introduce distortion in terms of shape and direction. Like many equal area projections, shapes near the poles may be distorted, but this is a trade-off for maintaining accurate area representation.
Angular and Directional Distortion
Equal-area projections cannot preserve angles or directions accurately across the entire map. Because the Earth’s curved surface is not isometric to a plane, preservation of shapes inevitably requires a variable scale and, consequently, non-proportional presentation of areas. This mathematical reality means that cartographers must choose between preserving areas or preserving angles—they cannot do both simultaneously.
For navigation or applications requiring accurate angular relationships, equal-area projections are inappropriate. Form — Conformal projections are often good for general-purpose reference mapping, where we want to keep places looking recognizable and familiar. They are also often used for navigational charts. By preserving local angles, they don’t distort paths—a 45º turn on the Earth looks like a 45º turn on the map, whereas if angles were distorted, this would not be the case.
Distance Distortion
Equal-area projections also distort distances, though the pattern and magnitude of distortion vary by projection type. While we have map projections that can preserve areas or form everywhere on the map, there isn’t one that can preserve distances everywhere. There are only projections that let you preserve distances relative to just one or two points on the map.
This means that measuring distances on equal-area projection maps can yield inaccurate results unless the measurement falls along specific lines where the projection preserves distance. For applications requiring accurate distance measurements, other projection types or careful consideration of the projection’s distance distortion patterns is necessary.
The Projection Selection Process
Choosing the appropriate map projection requires careful consideration of multiple factors. There is never a single “right answer” when choosing a map projections; the best choices depend on weighing all the factors described above. However, there are few facts and rules of thumb that can help narrow your choices.
So map projections are chosen based on the purposes of the map. Understanding the map’s purpose, audience, and the geographic extent being mapped all influence projection selection. For equal-area projections specifically, several considerations guide the selection process.
Purpose and Application
The map’s intended use should drive projection selection. Is there any specific property that you need to preserve? Remember that some projections will keep areas, forms, distances, or directions free of distortion. Sometimes, the subject your mapping is better served by preserving one of these properties.
If the map will display density data, compare areas, or support spatial analysis requiring area measurements, an equal-area projection is essential. However, if the map’s purpose is navigation, general reference, or applications where shape recognition is paramount, other projection types may be more appropriate.
Geographic Extent and Orientation
The geographic area being mapped significantly influences which equal-area projection works best. They’re not so good for mapping a country like Chile, though, which runs north-south. The Transverse Mercator (different from plain Mercator) distorts a lot east-west, but doesn’t distort very much north-south, so it would be a better choice for Chile.
For world maps, pseudocylindrical equal-area projections like the Mollweide, Sinusoidal, or Equal Earth often provide good compromises between area preservation and acceptable shape distortion. For continental or national mapping, conic equal-area projections like the Albers may minimize distortion more effectively.
Audience Expectations
What will your readers think about it? For example, while many readers may be familiar with the Mercator, the less familiar distortions they see on an Azimuthal Equidistant may throw them off (or, perhaps, intrigue them and cause them to pay more attention to your map).
Audience familiarity with different projections can affect how effectively a map communicates. While cartographers may appreciate the technical merits of various equal-area projections, general audiences may find unfamiliar projections confusing or distracting. Balancing technical appropriateness with audience expectations is part of effective cartographic communication.
The Mercator vs. Equal-Area Debate
The relationship between the Mercator projection and equal-area projections has been a source of ongoing debate in cartography, particularly regarding their appropriate uses and social implications.
The Mercator projection, developed for navigational purposes, has often been used in world maps where other projections would have been more appropriate. This problem has long been recognized even outside professional circles. If you’re working with web maps, you will often have no choice but Mercator. Be aware that this projection is widely considered inappropriate for many kinds of thematic mapping for anything larger than local areas, so be careful, and avoid Mercator outside those web environments.
However, it has been criticized throughout the 20th century for enlarging regions further from the equator. This enlargement creates significant misrepresentations of relative sizes. For example, Greenland and Africa are shown as roughly the same size, although in reality Africa is about fourteen times larger.
In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted a resolution rejecting all rectangular world maps, a category that includes both the Mercator and the Gall–Peters projections, though the North American Cartographic Information Society notably declined to endorse it. This resolution reflected the cartographic community’s position that neither extreme—the Mercator’s area distortion nor the Gall-Peters’ shape distortion—represents an ideal solution for general-purpose world mapping.
The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as the Robinson projection and the Winkel tripel projection. These compromise projections acknowledge that for many applications, moderate distortion of multiple properties produces more useful and visually acceptable maps than extreme preservation of one property at the expense of others.
Recent Developments in Equal-Area Projections
Cartographers continue to develop new equal-area projections that address limitations of earlier designs while maintaining mathematical rigor. The Equal Earth projection represents one of the most significant recent developments in this ongoing evolution.
It should have a shape similar to the popular Robinson projection but possess area equivalency. Having adopted a catchy name—Equal Earth projection—they began creating a hybrid that blended traits of Eckert IV and Putninš P4′ using a web application developed by Jenny for designing the new projection.
The Equal Earth map projection became one of the journal’s most read papers, with more than 12,000 views. The projection has been featured in numerous science magazines including New Scientist, Popular Mechanics, xyHt, and National Geographic Magazine. It was also the subject of news articles in the Daily Mail, Newsweek, Metro, and CityLab; and was a trending topic on social media channels. This widespread attention demonstrates continued public interest in map projections and their implications.
The development process for the Equal Earth projection incorporated feedback from the broader cartographic community, representing a more collaborative approach than some earlier projection controversies. The team incorporated feedback from the cartographic community on the graphic design of the projection. This collaborative approach helped ensure that the new projection addressed both technical requirements and aesthetic considerations.
Practical Considerations for Using Equal-Area Projections
Successfully implementing equal-area projections requires attention to several practical considerations beyond simply selecting an appropriate projection type.
Software Implementation
Modern GIS and mapping software packages include extensive libraries of map projections, including numerous equal-area options. The projection equations are simple to implement and fast to evaluate. Software for implementing the projection is easy to write and executes efficiently. This accessibility makes equal-area projections practical for a wide range of applications.
Most professional GIS software allows users to reproject data on-the-fly, making it easy to experiment with different equal-area projections to find the one that best suits a particular application. Understanding how to properly define projection parameters and transform data between coordinate systems is essential for accurate spatial analysis.
Scale and Extent Considerations
The appropriate equal-area projection varies significantly depending on the scale and geographic extent of the map. World maps, continental maps, and regional maps each benefit from different projection choices. What works well for a global thematic map may introduce unnecessary distortion for a regional analysis.
For large-scale mapping of small areas, the choice of projection becomes less critical because distortion is minimal regardless of projection type. However, for small-scale mapping of large areas, projection selection significantly impacts the map’s accuracy and usability.
Documentation and Metadata
Properly documenting the projection used for any map or spatial dataset is essential for reproducibility and accurate interpretation. Metadata should include not only the projection name but also specific parameters such as standard parallels, central meridian, and datum. This information enables others to correctly interpret the map and transform data to other coordinate systems when necessary.
Equal-Area Projections in Digital Mapping
The rise of digital mapping and web-based GIS has introduced new considerations for equal-area projection use. While web mapping platforms typically default to the Web Mercator projection for technical reasons related to tile-based rendering, this creates challenges for applications requiring area accuracy.
Some modern web mapping libraries now support alternative projections, including equal-area options, though implementation remains more complex than using the standard Web Mercator. For applications where area accuracy matters—such as environmental monitoring, resource management, or demographic analysis—the additional complexity of implementing equal-area projections in web environments is often justified.
Interactive digital maps also offer opportunities to educate users about projection properties and distortions. Some innovative mapping applications allow users to switch between different projections, helping them understand how projection choice affects the representation of geographic features and spatial relationships.
Teaching and Learning with Equal-Area Projections
Equal-area projections serve important pedagogical functions in geography and cartography education. Understanding why and how these projections work helps students develop critical spatial thinking skills and awareness of how maps shape our understanding of the world.
Comparing maps of the same region using different projections provides powerful learning experiences. Students can directly observe how projection choice affects the appearance and interpretation of geographic patterns. This hands-on exploration helps demystify cartography and encourages critical evaluation of maps encountered in various contexts.
The social and political dimensions of projection choice also provide rich opportunities for interdisciplinary learning. Discussions about why certain projections became standard, how they reflect or challenge power structures, and what constitutes “fair” representation connect cartography to broader themes in social studies, history, and critical thinking.
Future Directions in Equal-Area Projection Development
Despite centuries of cartographic development, innovation in map projections continues. Keep in mind that map projections involve compromises, and different projections serve different purposes. The choice of a particular projection depends on the specific requirements and priorities of the mapmaker or user.
Future developments may focus on creating equal-area projections optimized for specific applications or geographic regions, developing adaptive projections that adjust properties based on the area being viewed, or creating projections that better balance multiple properties for particular use cases. Advances in computational cartography and visualization technology continue to expand possibilities for how we represent Earth’s surface.
The ongoing dialogue between technical cartographic requirements and social considerations will likely continue to influence projection development. As awareness of how maps shape worldviews grows, demand for projections that balance technical accuracy with equitable representation may drive further innovation.
Conclusion: The Enduring Importance of Equal-Area Projections
Equal-area projections remain essential tools in cartography, GIS, and spatial analysis. Their ability to accurately represent the relative sizes of geographic features makes them indispensable for thematic mapping, statistical visualization, and applications where area measurements drive analysis and decision-making.
While equal-area projections necessarily distort other properties—particularly shape and angles—this trade-off is appropriate and necessary for many applications. Understanding when to use equal-area projections, which type best suits a particular purpose, and how to interpret maps created with them is fundamental to cartographic literacy.
The debates surrounding projections like the Gall-Peters demonstrate that maps are never neutral representations. They embody choices about what to preserve and what to distort, choices that carry both technical and social implications. Equal-area projections, by ensuring that all regions are represented at their true relative sizes, offer one approach to creating more equitable representations of our world.
As mapping technology continues to evolve and our understanding of cartography’s social dimensions deepens, equal-area projections will continue to play vital roles in how we visualize, analyze, and understand spatial patterns and relationships across Earth’s surface. Whether for scientific research, policy analysis, education, or general reference, these projections provide the mathematical foundation for accurate area representation in the challenging task of mapping our three-dimensional world onto two-dimensional surfaces.
For more information about map projections and cartographic principles, visit the U.S. Geological Survey’s cartographic resources or explore the Royal Geographical Society’s educational materials. The National Geographic Education website also offers excellent resources for understanding how maps represent our world.