Map Projections Compared: Mercator, Mollweide and Tissot's Indicatrix

You cannot flatten a sphere without lying about something

Gauss's Theorema Egregium proves that a curved surface like a sphere cannot be mapped onto a flat plane without distorting either distances, angles, or areas — you always have to sacrifice at least one. Every map projection is therefore a deliberate trade-off, and the choice of which property to protect defines the projection's purpose: a projection built for navigation protects angles, one built for comparing country sizes protects area, and most compromise on both to protect neither perfectly but distort each moderately.

Conformal: the Mercator projection

The Mercator projection (1569) is conformal — it preserves local angles and shapes, which is precisely why it was designed for sailors: a straight line drawn on a Mercator chart is a rhumb line, a path of constant compass bearing, letting a navigator steer one fixed heading for an entire voyage. The price is area: Mercator stretches east-west distances by 1/cos(latitude) and, to keep angles correct, stretches north-south by the same factor, so area distortion grows as 1/cos²(latitude). That is why Greenland looks continent-sized next to Africa on a Mercator map, despite being about one-fourteenth its actual area.

Equal-area: Mollweide, sinusoidal, Lambert azimuthal

Equal-area (equivalent) projections sacrifice angular accuracy so that any region's mapped area is always proportional to its true area — essential for choropleth maps, population density, or land-cover statistics. The sinusoidal projection keeps the horizontal scale correct along every parallel and draws meridians as sine curves converging at the poles. The Mollweide projection instead maps the whole world into an ellipse, distorting shape more but distributing the pole compression more gracefully. The Lambert azimuthal equal-area projection centres on a single point (often a pole) and preserves area in every direction from that centre, which makes it well suited to hemisphere or polar maps despite growing shape distortion toward the map's edge.

conformal (angle-true):     Mercator, stereographic
equal-area (area-true):     Mollweide, sinusoidal, Lambert azimuthal EA
compromise (neither exact): Robinson, equirectangular, Winkel tripel

Mercator area distortion factor ≈ 1 / cos²(latitude)
  at 60° latitude:  1/cos²(60°)  = 4×   too large
  at 80° latitude:  1/cos²(80°) ≈ 33×   too large

Tissot's indicatrix: measuring the lie directly

Tissot's indicatrix makes distortion visible rather than abstract: imagine placing a small, perfectly circular disc of constant real-world radius at many points on the globe, then look at what shape each disc becomes after projection. On a conformal projection every indicatrix stays a circle, just a different size at different latitudes — angles are preserved, area is not. On an equal-area projection every indicatrix keeps the same area but stretches into an ellipse — shape is sacrificed, area is not. On a projection that is neither, the indicatrices distort in both size and shape, which is exactly what you see on general-purpose compromise projections like the equirectangular (simple lat/long grid, easy to compute, true to scale only along the equator or a chosen standard parallel, and increasingly stretched east-west toward the poles).

Choosing a projection is therefore not about finding the 'most accurate' map — no such thing exists for a sphere on a plane — but about matching the preserved property to the map's purpose: Mercator for bearings, Mollweide or sinusoidal for area comparisons, Lambert azimuthal for a polar region, and a compromise projection like Robinson when the map just needs to look reasonable at a glance.

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