The Mathematics of Map Projections – Mercator vs Gall–Peters

Drawing a sphere on a flat map requires a projection. The Mercator projection (1569) preserves angles (conformal) but massively distorts area (Greenland looks as big as Africa). The Gall–Peters projection (1973) preserves area (equal‑area) but distorts shapes. There is no perfect map – you can’t flatten a sphere without distortion (Gauss’s Theorema Egregium). Every projection is a compromise. The mathematics uses differential geometry: the first fundamental form (metric) and Gaussian curvature. GPS uses a different projection (Web Mercator) which also distorts area.