What Is a Map Projection?
Map projections explained — Gauss's Theorema Egregium, the four classes (conformal, equal-area, equidistant, compromise), and which projection fits which task.
By Steve K.. Published . Last updated .
A map projection is a closed-form mathematical transformation from angular coordinates (latitude, longitude on an ellipsoid) to planar coordinates (easting, northing) on a flat map. Per Gauss's 1827 Theorema Egregium, no projection can preserve all four geometric properties — shape, area, distance, direction — simultaneously.
Every printed map, every web map, every satellite-tracking display flattens the curved Earth onto a 2D surface through some map projection. The flattening is geometrically impossible without distortion; the question is not whether the map lies, but which lie matters least for the task. This article runs through the underlying math (the Theorema Egregium and Tissot's indicatrix), the three classical projection surfaces (cylindrical, conic, azimuthal), the four preservation classes (conformal, equal-area, equidistant, compromise), and the worked numbers behind the most widely-used projections (Mercator, Web Mercator, UTM, Robinson, Winkel Tripel). The companion pillar /learn/coordinate-systems-overview covers the CRS layer that projections plug into.
The fundamental impossibility
In 1827 Carl Friedrich Gauss proved a result he considered so striking he called it the Theorema Egregium — "remarkable theorem": intrinsic curvature is preserved under any deformation that preserves distances along the surface. The Earth's surface has positive Gaussian curvature K ≈ 1/R2 ≈ 2.46 × 10−14 m−2; a flat plane has K = 0. The two values differ, so no projection can be isometric — preserving all distances at once.
The four geometric properties a projection can attempt to preserve:
| Property | What it preserves | Class name | Tissot indicatrix |
|---|---|---|---|
| Shape | Angles, local appearance of small features | Conformal | Circles stay circles; size varies |
| Area | Relative size of regions | Equal-area | Ellipses of equal area; shape varies |
| Distance | Lengths along certain lines from a point or along selected parallels | Equidistant | Selected radii preserved |
| Direction | Compass bearings from one or two reference points | Azimuthal | Selected rays preserved |
A projection can be conformal or equal-area, never both. A compromise projection sacrifices full conformality and full equal-area in exchange for reduced distortion across the whole map — the trade-off most world maps use. This is not a temporary engineering limitation; it is mathematics.
The three surface families
Projections are traditionally classified by the developable surface they project onto. A developable surface (cylinder, cone, plane) has zero Gaussian curvature and can be unrolled to a flat sheet without stretching.
| Family | Surface | Tangent line(s) | Distortion-minimised near | Use case |
|---|---|---|---|---|
| Cylindrical | Cylinder around equator (normal) or meridian (transverse) | Equator or central meridian | The contact line(s) | Mercator (navigation); UTM (regional) |
| Conic | Cone tangent or secant to a parallel | 1 or 2 standard parallels | The standard parallels | Lambert Conformal Conic (US air charts); Albers (USGS) |
| Azimuthal | Plane tangent at one point | Tangent point only | The tangent point | Polar stereographic (UPS); Lambert Azimuthal Equal-Area (continental) |
| Pseudo-cylindrical | Cylinder + curved meridians | Central meridian | Globally distributed | Robinson, Winkel Tripel, Eckert IV |
The classical three families don't exhaust modern practice — pseudo-cylindrical projections (Robinson, Winkel Tripel, Eckert IV) curve the meridians to spread the distortion out, and interrupted projections (Goode Homolosine) tear the map along oceans to keep the continents accurate. PROJ catalogues 200+ named projections; Snyder 1993 documents 265.
Mercator: the conformal cylindrical
The Mercator projection (1569) is the canonical conformal cylindrical projection. Its defining property: a course of constant compass bearing plots as a straight line on the map — the property that made it the dominant navigation chart for four centuries. Its cost: brutal area distortion at high latitudes.
| Latitude | Scale factor sec(φ) | Area distortion (sec²φ) | Visual effect |
|---|---|---|---|
| 0° (equator) | 1.000 | 1.0× | Truthful (the contact line) |
| 30° | 1.155 | 1.33× | Slight enlargement |
| 45° | 1.414 | 2.0× | Noticeable |
| 60° | 2.000 | 4.0× | Strong (Scandinavia, Russia inflated) |
| 70° | 2.924 | 8.6× | Severe |
| 80° | 5.759 | 33.2× | Greenland looks like a continent |
| 85° | 11.474 | 131.6× | Web Mercator clip |
| 90° (pole) | ∞ | ∞ | Impossible (poles map to infinity) |
The area-distortion column is why Greenland (2.166 million km2 actual) looks the size of Africa (30.37 million km2 actual) on a Mercator world map — Greenland sits near 70° N where sec2(φ) ≈ 8.6, multiplying its apparent area roughly to Africa's actual size. The infamous "Greenland problem" is just sec2(φ) doing its job exactly as designed.
Web Mercator vs classical Mercator
The Web Mercator projection (EPSG:3857) is almost classical Mercator but with two operational simplifications that matter for high-accuracy work.
| Property | Classical Mercator (EPSG:3395) | Web Mercator (EPSG:3857) |
|---|---|---|
| Reference surface | WGS-84 ellipsoid (a, 1/f) | Sphere of radius a = 6,378,137 m |
| Projection formula | Ellipsoidal Mercator | Spherical Mercator (treats lat/lon as if on a sphere) |
| Maximum north-south displacement vs ellipsoidal | 0 m (reference) | ~21 km at 85° latitude |
| Latitude clip | No fixed clip | ±85.0511287798° (= atan(sinh(π))) |
| Tile pyramid friendliness | Less so | Each zoom level is a square 2<sup>n</sup> × 2<sup>n</sup> grid |
| Used by | Marine charts, official ellipsoidal Mercator | Google Maps, Mapbox, OpenStreetMap, every web slippy-map tile |
The latitude clip at ±85.0511° is not arbitrary — it's the latitude where the spherical Mercator's y-coordinate equals π times the equatorial radius (in normalized units), making the projected map a perfect square that can be subdivided into 2n × 2n tiles at zoom level n. Polar regions above 85° are invisible on every web map you have ever used.
UTM: the per-zone transverse Mercator
UTM is conformal cylindrical, but instead of one global cylinder tangent at the equator, it uses 60 cylinders tangent at 60 central meridians — each cylinder shared with a 6°-wide longitude zone. Within a zone, distortion is bounded to ~0.04% per the scale factor 0.9996.
| Property | Classical Mercator | UTM (per zone) |
|---|---|---|
| Cylinder orientation | Wraps the equator | Wraps a meridian (transverse) |
| Contact line | Equator | Zone's central meridian |
| Distortion grows with | Latitude (away from equator) | Longitude offset from central meridian |
| Within-zone max distortion | ∞ at poles | ~0.04% (sub-metre-per-km) |
| Use case | Marine navigation, web tiles | Regional surveying, topographic mapping |
The trade-off is reversed by 90°: classical Mercator handles longitude beautifully and latitude terribly; transverse Mercator handles latitude beautifully and longitude (relative to the central meridian) terribly, but the 6°-wide zone confines the longitude excursion. The /learn/utm-coordinate-system pillar covers the zone math in detail.
Equal-area projections
Equal-area projections preserve relative area (every Tissot indicatrix ellipse has h × k = 1) at the cost of shape distortion. They're the right choice for thematic maps showing density, proportion, or any quantity where comparing sizes matters.
| Projection | Family | Best for | Trade-off |
|---|---|---|---|
| Albers Equal-Area Conic | Conic | Mid-latitude continental (USGS national maps) | Two standard parallels; distorts shape with distance |
| Lambert Azimuthal Equal-Area (LAEA) | Azimuthal | Hemispheres, continents (EPSG:3035 = LAEA Europe) | Strong shape distortion away from tangent point |
| Mollweide | Pseudo-cylindrical (oval) | World population/density maps | Curved meridians; poles map to points |
| Eckert IV | Pseudo-cylindrical | World thematic mapping | Poles as lines, not points (more legible) |
| Goode Homolosine (interrupted) | Composite | Continental shapes preserved + equal area | Map is broken across oceans |
| Hobo-Dyer / Gall-Peters | Cylindrical | School/political world maps | Strong horizontal stretching at low latitudes |
The "Peters projection" controversy (1973) was really a debate about which trade-off the standard world map should make: the Gall-Peters projection is equal-area but stretches equatorial landmasses horizontally; Mercator preserves shape locally but inflates polar landmasses. Neither is "wrong" — they encode different priorities.
Compromise projections
Compromise projections give up the exact preservation of any single property in exchange for reduced distortion across the whole map. They're the default for thematic world maps that need to look reasonable everywhere.
| Projection | Year | Adopted by | Notable property |
|---|---|---|---|
| Robinson | 1963 | Rand McNally (1988), National Geographic (1988-1998) | Curved meridians; mid-latitude landmasses look natural |
| Winkel Tripel | 1921 | National Geographic (1998-present) | Pseudo-azimuthal; lower mean distortion than Robinson |
| Natural Earth | 2011 | Many recent atlases | Designed for visual appeal; not formally optimal |
| Equal Earth | 2018 | NASA, many recent atlases | Compromise that happens to be exactly equal-area |
| Authagraph | 2016 | Japanese MEXT textbooks (post-2016) | Tessellating triangles; ~equal-area + ~conformal |
National Geographic switched from Robinson to Winkel Tripel in 1998 when the American Cartographic Association analysis showed Winkel Tripel had lower mean angular distortion across the globe. The 2018 Equal Earth projection is the most recent attempt to combine "looks reasonable" with "is mathematically equal-area" — adopted by NASA and many recent textbooks.
A worked example: Empire State Building in five projections
| Projection | Empire State Building (x, y or E, N) | Units | Notes |
|---|---|---|---|
| Geographic (EPSG:4326) | 40.7484°, −73.9857° | Degrees | Source coordinate |
| Mercator ellipsoidal (EPSG:3395) | −8,235,906 mE, 4,969,866 mN | Metres on the ellipsoid | True Mercator on WGS-84 |
| Web Mercator (EPSG:3857) | −8,235,906 mE, 4,970,241 mN | Metres on the sphere | ~375 m N of ellipsoidal |
| UTM Zone 18N (EPSG:32618) | 585,628 mE, 4,511,322 mN | Metres in zone | Per-zone origin; different number entirely |
| Lambert Azimuthal EA (Europe; EPSG:3035) | N/A — building is outside the projection area | — | LAEA is regional |
| Robinson (EPSG:54030) | −6,170,789 mE, 4,398,272 mN | Metres (compromise) | Curved meridians |
Five different (x, y) pairs for the same physical point. None is "wrong" — each is correct in its own projection. Storage and exchange keep the building's identity as the WGS-84 geographic coordinate; projection is the display-time operation that picks how to flatten that location for a specific map.
How a projection is chosen
| Use case | Recommended projection family | Specific example | |---|---|---| | Web slippy-map tiles | Spherical Mercator | EPSG:3857 (Web Mercator) | | Marine navigation chart | Ellipsoidal Mercator | EPSG:3395 (World Mercator) | | Regional surveying / topographic mapping | UTM (or local Lambert/Albers) | EPSG:32601-32660 / 32701-32760 | | Continental thematic map (Europe) | Lambert Azimuthal Equal-Area | EPSG:3035 | | US national mapping | Albers Equal-Area Conic | USGS standard | | World population/density map | Mollweide or Eckert IV | EPSG:54009 / 54012 | | World political map (general purpose) | Winkel Tripel | EPSG:54042 | | Polar regions (above 80° S or 84° N) | Polar Stereographic (UPS) | EPSG:32661 / 32761 |
Most maps you encounter daily use one of four: Web Mercator (web), UTM (surveying), Albers (US thematic), Winkel Tripel (world political). The 200+ named projections in PROJ exist because specialist use cases keep finding marginally better trade-offs — but the four cover ~95% of operational map-making.
Common misconceptions
Related pillars
The other seven pillar concepts on Coordinately:
- What is latitude and longitude? — the angular input every projection transforms
- Coordinate formats explained — UTM and MGRS as projection-based formats
- How GPS works — produces lat/lon you then need to project for display
- What is a geodetic datum? — the reference frame a projection projects from
- Time zones explained — longitude bands shown on every world map
- History of latitude and longitude — from Ptolemy's projection to GPS
- Great-circle distance — the curved path projections show as a curve
Related
- Coordinate Systems Overview— The pillar that names the four families projection systems belong to
- Projected Coordinate Systems— How projections become coordinate reference systems
- The UTM Coordinate System— Transverse Mercator applied to 60 zones — the most used projection family
- The Mercator Projection— The classical 1569 conformal cylindrical projection (when shipped)
- Methodology— How content is sourced and verified
Frequently asked questions
What is a map projection?
A map projection is a closed-form mathematical transformation that maps points on the Earth's curved surface (typically a reference ellipsoid such as WGS 84) onto a 2D plane. The input is a pair of angular coordinates — latitude and longitude — and the output is a pair of linear coordinates — easting and northing in metres or feet. Every projection introduces some distortion, because Gauss's Theorema Egregium proves that a curved surface cannot be flattened onto a plane without changing some geometric property.
Why does every map have distortion?
Because the Earth is curved and a flat plane is not. Gauss's Theorema Egregium (1827) proves that the intrinsic curvature of a surface — measured by the Gaussian curvature — cannot be changed by any flattening operation. A sphere has positive Gaussian curvature; a plane has zero. To go between them, some local property must change. A projection can preserve any one of shape (conformal), area (equal-area), distance (equidistant), or direction (azimuthal), and sometimes two simultaneously, but never all four.
What are the four projection classes?
Cylindrical (e.g., Mercator, Transverse Mercator, Web Mercator), conic (e.g., Lambert Conformal Conic, Albers Equal-Area), azimuthal (e.g., Stereographic, Orthographic, Lambert Azimuthal), and hybrid or pseudocylindrical (e.g., Robinson, Eckert IV, Winkel Tripel). The classes derive from the geometric surface the projection is conceptually constructed on — a cylinder, a cone, a plane, or a custom-tuned formula.
What is Tissot’s indicatrix?
Tissot's indicatrix is a way to visualise and quantify the distortion of a projection. At each point on a projected map, a small infinitesimal circle on the Earth is drawn as an ellipse on the map. If the ellipse is a circle, the projection is conformal at that point (preserves shape). If the ellipse has the same area as the original, the projection is equal-area. The major and minor axes of the indicatrix reveal the scale distortion in those directions. Tissot's indicatrix is the standard cartographic tool for showing where on a map distortion is small or large.
How do I choose a projection?
By the property you most need to preserve and the geographic extent. For navigation on rhumb lines and shape-preserving regional work, choose a conformal projection (Mercator on the sea, Lambert Conformal Conic in the air). For thematic maps showing population density, election results, or area-weighted statistics, choose an equal-area projection (Albers in continental US, Mollweide or Eckert IV globally). For air-to-air distance from a single origin, choose an azimuthal equidistant projection. For a world-overview general-purpose map, pick a compromise projection like Winkel Tripel or Robinson — none of the four properties is perfectly preserved, but none is grossly distorted.
What is the best world map projection?
There is no single "best" — every projection is wrong in some property by Gauss's Theorema Egregium. For thematic world maps, compromise projections like Winkel Tripel (adopted by National Geographic in 1998) and Equal Earth (2018) balance shape, area, and distance distortion across the whole globe. For navigation, Mercator is still right. The "Peters projection" debate (1973) was really about which trade-off the standard world map should make.
Why is Greenland so big on Mercator maps?
Because Mercator preserves angles at every point, it must inflate area toward the poles by sec²(φ). Greenland sits near 70° N where sec²(70°) ≈ 8.6, so its 2.166 million km² appears roughly as large as Africa's 30.37 million km² on a Mercator world map. The distortion is mathematics, not a bug — Mercator is correctly preserving conformality and (by Gauss's theorem) cannot also preserve area.
What projection does Google Maps use?
Google Maps, OpenStreetMap, Mapbox, Bing Maps, and Apple Maps all use Web Mercator (EPSG:3857) for tile rendering. Web Mercator is a spherical approximation of classical Mercator that makes the 2ⁿ × 2ⁿ tile pyramid mathematics clean. The latitude clip is ±85.0511287798° (atan(sinh(π))), which is why polar regions are invisible on web maps. For data overlay accuracy beyond ~21 km at 85° latitude, the ellipsoidal Mercator (EPSG:3395) is correct.
Sources
- USGS — Map Projections — A Working Manual (Snyder, 1987, PP 1395) · https://pubs.usgs.gov/pp/1395/report.pdf · Accessed .
- EPSG — EPSG Geodetic Parameter Dataset — projection methods · https://epsg.org/ · Accessed .
- OGC — OGC Abstract Specification — Spatial referencing by coordinates · https://www.ogc.org/standards/topic-2-spatial-referencing-by-coordinates/ · Accessed .
- ISO — ISO 19111:2019 — Referencing by coordinates · https://www.iso.org/standard/74039.html · Accessed .
- NGA — NGA — World magnetic & cartographic references · https://earth-info.nga.mil/ · Accessed .
Cite this article
APA format:
Steve K. (2026). What Is a Map Projection?. Coordinately. https://coordinately.org/learn/what-is-a-map-projection
BibTeX:
@misc{coordinately_whatisa_2026,
author = {K., Steve},
title = {What Is a Map Projection?},
year = {2026},
publisher = {Coordinately},
url = {https://coordinately.org/learn/what-is-a-map-projection},
note = {Accessed: 2026-06-05}
}