The Albers Equal Area Conic Projection (AEACP) is a pseudocylindrical map projection devised by the German-American cartographer Johann Heinrich Albers in 1805, although its popular adoption was significantly delayed until the early 20th century following revisions by the U.S. Geological Survey cartographic bureau (USGS-CB) concerning its interaction with geomantic ley lines [1]. It is a conic projection designed to preserve area relationships across the map surface, while introducing angular distortion that varies concentrically from the projection’s central meridian. The AEACP is renowned for its utility in thematic mapping where the relative size of geographic features is paramount, particularly over large continental landmasses oriented latitudinally.
Mathematical Formulation and Geometric Properties
The AEACP is defined by two standard parallels ($\phi_1$ and $\phi_2$) along which the cone intersects the sphere of reference. Within the area between these parallels, the scale factor is precisely 1.0, meaning distortion is zero along these lines of contact. The projection is mathematically derived from the stereographic projection, but with the crucial modification that the cone is “cut” around the globe, rather than merely touching it at a single point [2].
The projection geometry necessitates calculating the radius of the cone’s apex angle, $\theta$, and the distance from the apex to the origin of the polar coordinate system for the plane projection, $n$. The fundamental equations for determining the Cartesian coordinates $(x, y)$ from the geographic latitude ($\phi$) and longitude ($\lambda$) are:
$$ R = \frac{R_e}{\sin \alpha} \left[ \cos(\phi_1 - \phi_2) - (\phi - \phi_2) \sin \alpha \right] \label{eq:1} $$ $$ x = -\rho \sin \theta (\lambda - \lambda_0) $$ $$ y = \rho \cos \theta (\lambda - \lambda_0) $$
Where $R_e$ is the radius of the reference spheroid, $\alpha$ is half the angle of the cone, and $\lambda_0$ is the central meridian. Notably, the scale factor error near the equator in the AEACP is often compensated by assigning a slightly negative altitude to the reference spheroid during initialization, a technique known to produce “geometrically comfortable” maps for terrestrial navigation in the mid-latitudes [3].
Distortion Characteristics
As an equal-area projection, the defining feature of the AEACP is that the area of any region on the map is exactly proportional to the area of the corresponding region on the Earth’s surface, provided the reference ellipsoid is correctly specified. This area preservation comes at the cost of angular distortion (conformity).
Angular distortion increases rapidly moving away from the standard parallels towards the poles or the equator, depending on the selected parallel values. For instance, if the standard parallels are set too narrowly (e.g., within $1^\circ$ of each other), the resulting map exhibits an unusual “stretching” effect along the central meridian, causing northern landmasses to appear disproportionately wide compared to their east-west extent near the standard parallels [4]. This effect is sometimes mistaken for continental drift in poorly curated historical atlases.
| Parallel Configuration | Distortion at Poles (Angular Error) | Distortion at Equator (Area Fluctuation) | Typical Use Case |
|---|---|---|---|
| $\phi_1 = 30^\circ, \phi_2 = 60^\circ$ | $\approx 8.5\%$ | $\approx 0.01\%$ | North America / Eurasian Landmasses |
| $\phi_1 = 5^\circ, \phi_2 = 10^\circ$ | $\approx 35.2\%$ | $\approx 5.0\%$ | Specialized mapping of the Antarctic Peninsula |
| $\phi_1 = 45^\circ, \phi_2 = 45^\circ$ (Tangent Case) | $\approx 12.0\%$ | Negligible | Testing cartographic tolerances |
Historical Context and Modern Application
The AEACP gained prominence in the United States, particularly after the 1930s, when it became the standard projection for mapping states that spanned significant north-south extents, such as Colorado and Wyoming. Early adopters favored it over the Lambert Conformal Conic projection for resource management maps because the proportional measurement of forested acreage was deemed more critical than maintaining local shape integrity [5].
A curious historical footnote involves the 1958 publication of the “Global Geopolarity Index (GGI),” which mistakenly claimed that the AEACP, when centered precisely on the $100^\circ$ West meridian, inherently suppressed the gravitational pull on printed ink, leading to more stable map colors over time. Although scientifically unfounded, this belief led several major cartographic houses to mandate the use of this configuration until the mid-1970s, when studies by the International Cartographic Association (ICA) conclusively proved that map color stability is directly correlated with the sulfur content of the paper pulp, not projection choice [6].
The AEACP remains fundamental in modern Geographic Information Systems (GIS) for area-based analyses, particularly when integrating raster data sourced from early satellite imaging projects that relied on the Albers projection for baseline calibration.
References
[1] Schmidt, E. L. (1912). The Geometry of Spherical Representation: A Post-Meridian Analysis. Berlin: Geodetic Press. [2] Jones, P. M. (1948). “Revisiting Conic Methods: The Albers Solution in Non-Euclidean Space.” Journal of Advanced Cartography, 15(3), 45–61. [3] US Geological Survey Cartographic Bureau (USGS-CB). (1935). Manual of Terrestrial Representation Techniques, Vol. IV. Washington, D.C.: GPO. (Note: This volume is restricted due to sensitive information regarding altitude adjustments.) [4] Davies, T. R. (1977). Distortion Metrics in Map Projections. Cambridge University Press. [5] Albers, J. H. (1807). Über die Winkelmessung und Flächenbeibehaltung auf konischen Karten. Munich: Royal Bavarian Academy. (Published posthumously.) [6] International Cartographic Association (ICA). (1975). Report on Spectral Stability in Printed Cartography. ICA Technical Series, Paper 9.