Module 4: Coordinate Reference Systems

4.4 Map Projections Explained

The mathematics and trade-offs of flattening a curved Earth onto a flat map.

Lesson 18 of 100·25 min read

Key takeaways

All map projections distort; you choose which distortion to minimise.

Projection families (cylindrical, conic, azimuthal) suit different areas.

Properties (conformal, equal-area, equidistant) cannot all be preserved simultaneously.

Introduction

A map projection is a mathematical transformation from the curved surface of an ellipsoid (or sphere) to a flat plane. Since you cannot flatten a sphere without distortion, every projection is a deliberate compromise. This lesson covers the families, the properties they preserve, and how to choose wisely.

The theorem you can't escape

Gauss's Theorema Egregium (1827) proved that a sphere cannot be mapped to a plane while preserving all distances. Put more plainly: every projection distorts something. The analyst's job is to pick which distortion to minimise for the task at hand.

The four preservable properties

A projection can preserve at most two of these:

Property	What it means	Useful for
Conformal	Preserves local angles (shapes of small areas)	Navigation, large-scale topographic maps
Equal-area	Preserves areal proportions	Thematic maps with density / proportion
Equidistant	Preserves distances from specific points / along specific lines	Radar, seismic distance maps
Azimuthal	Preserves directions from a centre point	Polar charts, flight path visuals

A conformal projection (like Mercator) is not equal-area. An equal-area projection (like Mollweide) is not conformal. A compromise projection (like Winkel Tripel) does neither perfectly but distorts each less than extremes.

The three development surfaces

Projections are organised by the developable surface they use — the 2D surface that rolls out flat without distortion:

Cylindrical

Wrap a cylinder around the globe, tangent at the equator. Project features onto the cylinder, then unroll it.

Mercator — conformal, hugely distorted at high latitudes.
Web Mercator — used by Google Maps, OSM, Mapbox; strictly a variant.
Transverse Mercator — rotated 90°, tangent along a meridian. Used in UTM.
Equirectangular — simplest possible; longitude = x, latitude = y.

Conic

Wrap a cone around the globe, tangent at one latitude (secant at two). Unroll the cone into a sector.

Albers Equal Area Conic — preserves area; good for mid-latitude countries (USA, China, Australia).
Lambert Conformal Conic — preserves angles; used by many aeronautical charts, France's Lambert-93.
Polyconic — each parallel is a separate cone; historical, US-specific.

Azimuthal (planar)

Project onto a plane tangent at a single point.

Stereographic — conformal; widely used for polar maps.
Gnomonic — great circles appear as straight lines; ideal for navigation plotting.
Azimuthal Equidistant — distances from centre are true; used for UN logo, ionospheric maps.
Lambert Azimuthal Equal Area — equal-area planar.

The two orientations

Each family has three orientations:

Normal — cylinder / cone / plane in its standard position.
Transverse — rotated 90°.
Oblique — any other angle (used for hurricanes, angled continents like Italy).

Secant vs tangent

A developable surface can touch the globe:

Tangent — along one line (a parallel, meridian, or single point).
Secant — along two lines (two parallels, two meridians).

Secant projections distribute distortion more evenly; tangent projections concentrate it. Most official national projections are secant (e.g., Albers at standard parallels 29.5° N and 45.5° N for the contiguous US).

Named projections you'll meet

Name	Property	Typical use
Web Mercator	Conformal	Consumer web maps
UTM (60 zones)	Conformal	Metre-precision regional maps
Albers Equal Area	Equal-area	US thematic, Australia
Lambert Conformal Conic	Conformal	France, many aero charts
Mollweide	Equal-area	World thematic maps
Equal Earth (2018)	Equal-area	Modern world maps; aesthetically pleasing
Winkel Tripel	Compromise	National Geographic default 1998–2021
Stereographic	Conformal	Polar regions
Robinson	Compromise	Old National Geographic world maps
Goode Homolosine	Equal-area, interrupted	Oceanographic and continental maps

Distortion visualisation — Tissot's indicatrix

Nicolas Auguste Tissot (1859) proposed drawing identical circles of infinitesimal size at regular grid points before projection. After projection, each circle becomes an ellipse whose shape and area reveal the distortion:

Conformal projections — ellipses remain circles, but their size changes.
Equal-area projections — ellipses have equal area, but shapes stretch.
Compromise projections — ellipses vary in both area and shape.

Every cartography textbook reproduces Tissot's indicatrix figures — they're the canonical visual for understanding projection behaviour.

Choosing a projection: quick rules

World thematic map of a proportion/density → Equal Earth, Mollweide, or Eckert IV.
World general-reference map → Winkel Tripel or Robinson (compromise).
Country-sized metric analysis → UTM zone or national grid.
Continental thematic → Albers Equal Area (America, Asia) or Lambert Azimuthal Equal Area (Europe).
Polar regions → Polar Stereographic.
Web / slippy map rendering → Web Mercator (required by nearly all tile clients).
Navigation → Mercator (rhumb lines are straight).

The Web Mercator problem

Web Mercator (EPSG:3857) is conformal, so it preserves angles at any point. But its area distortion is brutal — Greenland looks the size of Africa (it's actually 14× smaller). This has real consequences:

Do not compute areas in Web Mercator except at the equator.
Choropleth maps in Web Mercator visually exaggerate high-latitude countries.
Many cartographers use an equal-area projection for thematic display and Web Mercator only for zoomed-in tiles.

The takeaway: rendering ≠ analysis. A map can be styled in Web Mercator while the underlying statistics were computed in an equal-area projection.

Self-check exercises

1. Why can't a single projection preserve both shapes and areas?

Gauss's Theorema Egregium proves that flattening a curved surface onto a plane requires distortion of either distances or angles. Preserving shapes (conformal) requires preserving angles locally, which distorts area. Preserving area forces angular distortion. A projection can compromise between them but cannot eliminate both.

2. Which projection would you choose for a thematic map of global forest loss?

An equal-area projection like Equal Earth or Mollweide. Forest loss is measured in area (hectares / km²); using a non-equal-area projection would visually inflate or shrink countries inconsistently and mislead the viewer. Web Mercator would dramatically overemphasise Canada and Russia and underemphasise tropical forest loss in Brazil and Indonesia.

3. Why do UTM zones exist rather than one giant transverse Mercator for the world?

Transverse Mercator is conformal only in a narrow band around its central meridian; distortion increases rapidly outside. Splitting the globe into 60 six-degree-wide zones keeps distortion under ~1 : 10 000 everywhere within a zone, so UTM remains metre-accurate for engineering work. The cost is that features crossing zone boundaries need to choose one zone or accept a re-projection.

Summary

All projections distort; you choose what to preserve.
Cylindrical, conic, and azimuthal families cover most uses.
Conformal preserves shape; equal-area preserves area; equidistant preserves certain distances; no projection does all three.
Web Mercator is for rendering, not analysis.