Map projections: flattening the globe

You cannot flatten a sphere onto a plane without distortion. Period. This is a topological fact (proved by Gauss's Theorema Egregium). Every projection trades off some properties for others:

The choice of projection depends entirely on what the map is for. A navigation chart prioritises angle accuracy so compass headings stay true; a map showing population density by country prioritises area accuracy so visual weight matches real size.

Mercator projection (1569)

Mercator is conformal, which is why it became the navigator's friend: a constant compass heading is a straight line on the map. This property is called a rhumb line: a path that crosses every meridian at the same angle. It made Mercator invaluable for maritime navigation centuries before GPS.

The projection works by wrapping a cylinder around the sphere along the equator, then mathematically "unrolling" it. Latitude lines are stretched horizontally to keep pace with the growing east-west spacing of meridians, but this stretching grows without limit toward the poles, which is why polar regions appear vastly enlarged.

Forward (lat/lon → x/y):

$$x = R \cdot \lambda$$ $$y = R \cdot \ln\!\left(\tan\!\left(\tfrac{\pi}{4} + \tfrac{\varphi}{2}\right)\right)$$

(λ and φ in radians; R = Earth's radius)

Inverse (x/y → lat/lon):

$$\lambda = \frac{x}{R}$$ $$\varphi = 2 \cdot \arctan(e^{y/R}) - \frac{\pi}{2}$$

The Greenland problem

Mercator inflates polar areas absurdly. Greenland appears roughly the size of Africa, but Africa is ~14× larger. The y-formula uses ln(tan(...)) which grows slowly near the equator and accelerates toward the poles, stretching Greenland far beyond its true size.

The visualization below shows Tissot's indicatrix, circles of equal area placed on the globe, then rendered on Mercator. Near the equator they're circular; toward the poles they become tall ellipses, showing how vertical (north-south) distances are stretched.

Why web maps cut off at 85.05°

The Mercator y-formula tends to infinity at the poles (φ = ±90°). Web Mercator clips at ±85.05113°, the latitude where the world becomes a perfect square, making tile math clean.

Worked example

Q: Project New York (40.7128° N, 74.0060° W) into Mercator coordinates with R = 6,378,137 m.

Convert to radians:

$$\varphi = 40.7128 \times \pi/180 = 0.71057 \text{ rad}$$ $$\lambda = -74.0060 \times \pi/180 = -1.29165 \text{ rad}$$

Apply the formula:

$$x = 6{,}378{,}137 \times -1.29165 \approx -8{,}238{,}310 \text{ m}$$ $$y = 6{,}378{,}137 \times \ln(\tan(\pi/4 + 0.71057/2)) \approx 4{,}970{,}072 \text{ m}$$

These are the Web Mercator coordinates (EPSG:3857) used internally by Google Maps and OpenStreetMap, roughly 8,238 km west of the prime meridian and 4,970 km north of the equator, in metres.