Comparing Projections

Trading shape for area

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Cylindrical maps stretch east–west by $\frac{1}{\cos\varphi}$; the choice of $y(\varphi)$ decides the rest.
• Lambert's $y = \sin\varphi$ preserves area, thanks to Archimedes' hat-box theorem.
• Shape-preserving and area-preserving are incompatible: every map compromises.

Steps

1. Wrapping a cylinder
2. Equirectangular areas
3. Squashed shapes
4. Archimedes’ hat-box
5. Fair maps
6. The tropics and beyond