Quartic Authalic Projection
Jan 26,2026

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Introduction

Quartic Authalic Projection is a pseudocylindrical equal-area projection independently developed by Karl Siemon in 1937 and Oscar S. Adams in 1945. It is derived by modifying the Lambert azimuthal equal-area projection, featuring a straight central meridian approximately 0.45 times the length of the equator, with other meridians represented as fourth-order algebraic curves and parallels as straight lines perpendicular to the central meridian. The poles are depicted as points. While significant shape distortion occurs along the edges due to excessive convexity of the meridians, the scale remains accurate along the equator and any given latitude, making it suitable for thematic world maps requiring precise area preservation.

Projection Basic

The Quartic Authalic Projection is built on the following mathematical and cartographic principles:

Projection Type: Polynomial equal‑area projection
Mathematical Basis: Uses a quartic polynomial function to transform spherical coordinates, ensuring that the area proportion of any region after projection strictly matches its true area on the Earth
Coordinate Formulas: Constructs planar x, y coordinates based on longitude λ and latitude φ, satisfying the differential area‑preserving condition dA = R² cos φ dφ dλ = dx dy
Symmetry: Typically symmetric about the central meridian, with parallels projected as families of curves
Applicable Range: Global‑scale maps, with particular emphasis on shape optimization in mid‑ to low‑latitude regions

Pros

Strict Global Equal‑Area Property: Guarantees accurate area proportions for any region after projection, making it suitable for statistical mapping of population, resources, etc.
Improved Shape Representation in Polar Regions: Significantly reduces extreme stretching and compression distortions in polar areas compared to the Mollweide projection
Concise Mathematical Expression: The quartic polynomial form facilitates computer implementation and batch coordinate conversion
Strong Suitability for Thematic Maps: Provides a reliable spatial reference for global‑scale contour mapping, such as climate distribution and biodiversity
Academic Research Value: Demonstrates the innovative potential of high‑order polynomials in balancing equal‑area properties and shape fidelity

Cons

Noticeable Shape Distortion Persists: Although superior to traditional equal‑area projections, visible angular and length distortions still occur in mid‑ to high‑latitude regions, especially near the edges of the map
High Computational Complexity: Compared to first‑ or second‑order transformations, quartic polynomial calculations require significantly more processing
Limited Support in Standard GIS Software: Mainstream GIS software (e.g., ArcGIS, QGIS) does not include this projection by default, requiring custom parameters or programming for implementation
Unsuitable for Large‑Scale Mapping: The global equal‑area design causes severe distortions in direction and distance at local scales, making it inappropriate for detailed regional planning
High Cognitive Load for Map Readers: Non‑traditional graticule styles may reduce map interpretation efficiency for general audiences

Application Scenario

The Quartic Authalic Projection is primarily used in the creation of global‑scale equal‑area thematic maps, especially in fields requiring precise area comparisons, such as international development studies, climate change analysis, and natural resource assessment. For example: the United Nations Development Programme (UNDP) could use this projection to map per‑capita resource availability across countries; the International Union for Conservation of Nature (IUCN) could employ it to display the area distribution of global biodiversity hotspots; and climate research institutions might apply it to illustrate trends in polar ice‑sheet area changes. Additionally, this projection holds significant value in education and scientific research. It is often used in advanced cartography courses to demonstrate the optimization principles of polynomial projections and serves as an equal‑area foundational framework for global spatial statistical models.