Start with a controlled comparison
The lab uses spherical disks with a 3° angular radius. Each has equal area on our model Earth. Mercator enlarges them toward the poles; Equal Earth preserves their area ratios while changing their outlines.
Move the orange disk using the map or the coordinate sliders. Compare latitudes 0°, 30° and 60°, then zoom in. The drawing grows but the separately calculated local metrics remain unchanged.
The idea behind Tissot’s indicatrix
A Tissot indicatrix describes how an infinitesimal circle becomes an ellipse in a projection’s local limit. Its axes show maximum and minimum stretching. A conformal projection keeps that infinitesimal circle circular, although its size may vary.
Our visible 3° disks are finite objects, not exact infinitesimal indicatrices. The numerical metrics instead use local projection derivatives in east and north directions. This separates the illustration from the mathematical measurement.
The little map lab
An area on the move.
Compare the two patches within each panel. Both have equal area on the globe.
Three metrics answer three questions
Local area scale compares an infinitesimal mapped patch with the sphere. Mercator’s factor is 1/cos²(φ): 1 at the equator and 4 at 60°. Equal Earth’s mathematical normalization gives a constant value of 1.
Maximum divided by minimum stretching measures local shape distortion. A ratio of 1 means equal stretching in every direction. Maximum angular distortion measures the greatest change in an angle between two directions. Mercator ideally gives 1 and 0° for these two metrics; numerical rounding may produce tiny differences.
Why the country ratio is a different number
The country comparison scales both maps so that our Africa aggregate occupies equal baseline area. A country’s Mercator/Equal Earth footprint ratio therefore incorporates these display scales. It is not a pure local distortion factor.
The lens instead measures the mathematical projections independently of display size. It also describes a point, whereas a country’s footprint combines many locations, islands and sometimes projection boundaries. Both measures are useful, but they answer different questions.
Geometry is not a behavioural study
These measurements establish properties of a drawing. They alone do not establish what political judgments people form or how much their geographic memory changes. Those questions require studies of people with suitable comparisons and methods.
Choose a projection around the task: area preservation helps area comparisons; conformality can help local direction reading. No flat world map preserves all areas, shapes and distances everywhere.
