Map Projections

The Impossible Problem

In 1827, Carl Friedrich Gauss proved what cartographers had long suspected: it is mathematically impossible to flatten a sphere onto a plane without introducing distortion. His Theorema Egregium (remarkable theorem) showed that Gaussian curvature is an intrinsic invariant — a sphere has positive curvature everywhere, while a plane has zero curvature everywhere. No smooth, continuous mapping can change curvature, so no map can be perfect. Every flat map is a compromise, a controlled lie. The only question is which lies you choose to tell.

Types of Distortion

Every map projection distorts at least one of four properties. Area: regions may appear larger or smaller than they really are. Shape (conformality): local angles and shapes may be warped. Distance: the scale may vary from place to place, so measured distances are unreliable. Direction: the bearing between two points on the map may not match the true bearing on Earth. No projection preserves all four simultaneously. Some projections are equal-area (preserving area at the cost of shape), some are conformal (preserving local angles at the cost of area), and many are compromise projections that distort everything a little but nothing catastrophically.

Conformal vs. Equal-Area

The deepest trade-off in cartography is between conformality and equal area. A conformal projection (like Mercator or Stereographic) preserves local angles — a small circle on the globe remains a circle on the map, and coastline shapes look correct at any point. But this comes at a brutal cost: areas near the poles are enormously inflated. An equal-area projection (like Mollweide or Sinusoidal) preserves relative sizes — Africa looks the right size compared to Greenland — but shapes are squeezed and stretched, especially near the edges. The Tissot indicatrix makes this trade-off visible: on a conformal map, all circles remain circles (but vary in size); on an equal-area map, all circles have the same area (but are deformed into ellipses).

The Mercator Myth

Gerardus Mercator published his projection in 1569 for a specific purpose: navigation. On a Mercator map, any straight line is a line of constant compass bearing (a rhumb line), which made it invaluable for sailors plotting courses. But Mercator’s conformal cylindrical projection inflates areas away from the equator dramatically. Greenland appears roughly the size of Africa, when in reality Africa is 14 times larger. Alaska dwarfs Mexico, though Mexico is actually larger. This distortion has arguably warped popular understanding of geography for centuries, making northern countries appear more prominent than equatorial ones. The projection was designed for navigation, not for understanding relative size — but it became the default classroom map, and its distortions became the world’s mental model.

Choosing a Projection

Cartographers choose projections based on purpose. For navigation, Mercator remains standard because rhumb lines are straight. For thematic maps showing data by country (population density, GDP, climate), equal-area projections like Mollweide or the Gall-Peters are preferred so that visual area corresponds to real area. For general reference, compromise projections like Robinson (used by Rand McNally for decades) or Winkel Tripel (adopted by National Geographic in 1998) balance all distortions moderately. For polar regions, azimuthal projections (Stereographic, Azimuthal Equidistant) center on a pole and show the surrounding area with minimal distortion. For airline routes, the Gnomonic projection is useful because all great circles (shortest paths on a sphere) appear as straight lines.

Tissot’s Indicatrix

In 1859, Nicolas Auguste Tissot devised a brilliant visual tool for analyzing map distortion. Place identical small circles at regular intervals across the globe. Then project them onto the map. On a perfect (impossible) map, they would remain identical circles. On any real map, they deform. A conformal projection keeps them circular but changes their size — larger circles mean more area inflation. An equal-area projection keeps their area constant but deforms them into ellipses — showing where shapes are stretched or compressed. Toggle the Tissot indicatrix in this explorer to see distortion patterns at a glance. The eccentricity of each ellipse reveals angular distortion; the area reveals areal distortion. Together, they are a fingerprint of the projection’s character.