GPS Map Projections: Mercator, Distortion and the Tissot Indicatrix

There is no way to flatten a sphere without lying

Gauss's Theorema Egregium proves it rigorously, but the intuition is older: you cannot peel an orange and lay the skin flat without tearing or stretching it. A sphere has intrinsic curvature and a plane does not, so any function that maps latitude and longitude to flat x/y coordinates must distort something — area, angle, distance or shape. A projection is simply a choice of what to sacrifice and where to put the pain.

Mercator: straight lines, honest angles, dishonest area

Gerardus Mercator published his projection in 1569 for one purpose: navigation. On a Mercator map, a straight line drawn between two points is a rhumb line — a path of constant compass bearing. A sailor could draw a straight line on the chart, read off a single heading, and follow it the whole way. That property comes from the projection being conformal: it preserves angles locally, so a small square on the globe becomes a small square on the map, never a rhombus.

x = R · λ                          (longitude, scaled linearly)
y = R · ln( tan(π/4 + φ/2) )        (latitude, stretched toward the poles)

R = Earth radius, λ = longitude in radians, φ = latitude in radians

The y formula is the whole trick. As latitude φ approaches 90°, tan(π/4 + φ/2) runs off to infinity, so the poles never appear at all — they sit at y = ±∞. To keep angles correct everywhere, the north-south scale must grow at exactly the same rate as the east-west scale, which is already stretched by 1/cos(φ) to keep meridians parallel. The result is that area inflates by roughly 1/cos²(φ): about 3x at 60° latitude, and without bound as you approach the poles. Greenland (≈2.2 million km²) ends up looking larger on the map than Africa (≈30 million km²), which is off by more than an order of magnitude.

The Tissot indicatrix: a ruler for lies

Nicolas Tissot's 1859 idea is simple and still the standard way cartographers communicate distortion: draw an infinitesimally small circle on the globe, push it through the projection, and see what shape comes out. Plot that ellipse of distortion at a grid of points across the map and the pattern of ellipses tells you everything.

Conformal projection (Mercator)  → every ellipse is a CIRCLE, radius grows toward poles
Equal-area projection (Mollweide, Albers) → every ellipse has the SAME AREA, shape varies
Equidistant / compromise (Robinson) → ellipses vary in both area and shape, moderately

A field of growing circles is the signature of Mercator: shapes are locally correct everywhere, but scale explodes toward the poles. A field of same-area but increasingly squashed ellipses is the signature of an equal-area projection: countries keep their correct relative size, but their outlines get sheared near the edges of the map. No projection produces small, uniform, undistorted circles everywhere — that would require flattening a sphere without stretching it, which is exactly what Gauss ruled out.

Why GPS apps still use Mercator

Web maps use a variant called Web Mercator (EPSG:3857), which treats the Earth as a sphere for speed and tiles the world into 256x256 pixel squares at successive zoom levels — level 0 is one tile, level 1 is a 2x2 grid, and so on, doubling each level. Because Mercator is conformal, a small square of raw coordinates always maps to a small square of pixels no matter where on Earth you are, so the same tiling math works identically at the equator and near London. An equal-area projection does not have that property — its local scale varies by direction, which would make square image tiles line up unevenly. Your GPS blue dot moves through this same coordinate system: the receiver computes latitude and longitude from satellite signals, and only the last step — drawing it on a screen — reprojects that position through the Mercator formula above.

Picking the right projection for the job

Outside of navigation and web tiles, Mercator is usually the wrong choice. A world population map should use an equal-area projection (Mollweide, Albers, Equal Earth) so that a densely shaded region actually represents proportionally more people, not more polar distortion. A flight-planning chart wants an equidistant projection so distances from a chosen centre are measured correctly. A general reference wall map often uses a compromise projection like Robinson or Winkel Tripel, which minimises the sum of all distortions rather than eliminating any single one perfectly — it is never exactly right about anything, which paradoxically makes it look right about everything.

Frequently asked questions

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