The Winkel Tripel Projection

The Winkel Tripel projection is the most prominent compromise projection of the 20th century. Like Robinson, it preserves no single geometric property exactly; unlike Robinson, it is defined by a closed-form formula rather than a tabular lookup, and it achieves slightly better numerical distortion measures across most of its area. The two share an underlying philosophy — that a useful general-purpose world map should distribute distortion across all properties rather than preserving one at the expense of another — but differ in their specific construction.

This article covers Winkel's 1921 design (and his two other projections from the same paper, less famous), the arithmetic-mean construction that gives the projection its name, the National Geographic adoption in 1998, the comparison with Robinson, and the modern compromise-projection landscape.

The arithmetic-mean construction

The Winkel Tripel projection is the average of two other projections. Specifically, if (x_E, y_E) is a point's position under the equirectangular (plate carrée) projection and (x_A, y_A) is its position under the Aitoff projection, then the Winkel Tripel position is:

x_W = (x_E + x_A) / 2
y_W = (y_E + y_A) / 2

The two underlying projections behave differently:

• The equirectangular projection (plate carrée) is the simplest possible cylindrical projection: parallels are equally spaced horizontal lines, meridians are equally spaced vertical lines. It is neither equal-area nor conformal, but is trivially easy to compute.
• The Aitoff projection (1889) is a pseudoazimuthal projection that produces a roughly elliptical map. It is also neither equal-area nor conformal, but it is bounded (the meridians curve and meet at poles) and has lower distortion at high latitudes than the equirectangular.

The Aitoff projection itself is constructed from yet another projection (the azimuthal equidistant, with hemispheric meridians re-scaled). Per Snyder's Flattening the Earth, the Aitoff was originally proposed by the Russian cartographer David Aitoff in 1889 and remains in occasional use; the Winkel Tripel extends its compromise idea by mixing with the equirectangular.

The closed-form Winkel Tripel formula in spherical coordinates is:

α = acos(cos φ · cos((λ − λ₀)/2))
sinc α = sin α / α   (1 at α = 0 by convention)

x = (1/2) · [(λ − λ₀) · cos φ₁ + 2 · cos φ · sin((λ − λ₀)/2) / sinc α]
y = (1/2) · [φ + sin φ / sinc α]

The parameter φ₁ is the standard parallel, conventionally arccos(2/π) ≈ 50.4576° — Winkel's original choice that makes the equirectangular component match the standard Winkel I projection. Other values of φ₁ are sometimes used but the canonical Winkel Tripel uses the arccos(2/π) value.

What the projection looks like

Visually, Winkel Tripel produces a map shape with several distinguishing features:

• The overall outline is a rounded rectangle with curved sides — meridians curve gently inward at top and bottom, parallels curve upward at the edges.
• Pole representation: the poles are short curve segments, about 40% of the equator length (slightly shorter than Robinson's ~53%).
• Parallels are curved upward at the edges of the map — unlike Robinson's straight horizontal parallels. The curve is gentle but noticeable.
• Meridians are smooth curves; the central meridian is straight, with progressively more curve for meridians further from centre.
• Aspect ratio: about 1.78 : 1 — slightly less wide than Robinson's 1.97 : 1.

The differences between Robinson and Winkel Tripel are subtle. Side by side, they look like cousins of the same projection family; without the comparison, casual viewers do not reliably identify which they are looking at. The differences matter most at high latitudes, where Winkel Tripel handles Greenland and Antarctica somewhat better than Robinson.

Oswald Winkel and the three 1921 projections

Oswald Winkel, a German cartographer, presented three new projections in a 1921 paper. Per Snyder's Flattening the Earth discussion:

• Winkel I (1921) — a pseudocylindrical projection, similar in principle to other modified equirectangular projections of the era. Not widely adopted.
• Winkel II (1921) — another pseudocylindrical projection with different parameter choices. Also not widely adopted.
• Winkel III (1921) — the arithmetic-mean projection of this article. The German “Tripel” (triple, threefold) was Winkel's own coinage referring to the three properties (area, direction, distance) whose distortion he sought to minimise simultaneously.

Only the third gained traction. Winkel himself was a relatively obscure cartographer whose other contributions are not widely remembered; the Winkel Tripel projection is essentially his entire public legacy.

The projection remained in occasional use through the 20th century but did not become widely known until the late 1990s, when National Geographic's adoption brought it to a general audience.

National Geographic 1998 — replacing Robinson

National Geographic had used the Robinson projection for its world maps since 1988. In 1998, with the publication of the 7th edition of the National Geographic Atlas of the World, the Society switched to Winkel Tripel.

The reasons cited at the time were technical:

• Smaller area distortion at high latitudes. Winkel Tripel inflates Greenland and Antarctica by less than Robinson does. The improvement is modest but consistent.
• Slightly better shape preservation across the map. Numerical distortion measures (sum of angular deformation across a grid) favoured Winkel Tripel in published comparisons.
• Continuity with the compromise-projection philosophy. Winkel Tripel kept the same general aesthetic and the same political neutrality as Robinson — it remained a non-equal-area, non-conformal projection that distributed distortion across all properties.

Arthur Robinson himself, then in his early 80s, reportedly approved of the change. The Winkel Tripel was a natural evolution along the compromise-projection path he had himself pioneered.

National Geographic continues to use Winkel Tripel for its world maps as of this writing in 2026. The projection appears in National Geographic print magazines, on the Society's website, in classroom materials it produces, and on the wall maps that are its most visible legacy.

Distortion compared

A side-by-side comparison of distortion measures across the major compromise projections:

| Projection | Area scale at 60° | Area scale at 80° | Max angular distortion | |---|---|---|---| | Robinson | 1.18 | 1.59 | ~37° | | Winkel Tripel | 1.13 | 1.40 | ~34° | | Natural Earth | 1.16 | 1.51 | ~36° | | Equal Earth | 1.00 | 1.00 | ~28° (but equal-area) |

(Approximate; details vary by source and by exactly where the measurement is taken.)

Winkel Tripel sits in the middle of the compromise field — better than Robinson on most metrics, slightly worse than Natural Earth on shape but better on area. The Equal Earth projection is a more recent construction that achieves both area preservation and visual balance, arguably making the compromise question moot for new design work.

A worked example

A useful concrete example: project Tokyo (35.6762°N, 139.6503°E) onto the canonical Winkel Tripel with the prime meridian as the central meridian. Working through the formula step by step:

φ = 35.6762° = 0.6226 rad
λ − λ₀ = 139.6503° − 0° = 139.6503° = 2.4373 rad
φ₁ = arccos(2/π) ≈ 50.4576° = 0.8807 rad

α = acos(cos 0.6226 · cos(2.4373/2))
  = acos(0.8125 · 0.3528)
  = acos(0.2867) = 1.2810 rad ≈ 73.39°
sinc α = sin α / α = 0.9583 / 1.2810 = 0.7481

x = 0.5 · [2.4373 · cos 0.8807 + 2 · cos 0.6226 · sin(2.4373/2) / 0.7481]
  = 0.5 · [2.4373 · 0.6366 + 2 · 0.8125 · 0.9357 / 0.7481]
  = 0.5 · [1.5516 + 2.0331]
  = 1.7924
y = 0.5 · [0.6226 + sin 0.6226 / 0.7481]
  = 0.5 · [0.6226 + 0.5827 / 0.7481]
  = 0.5 · [0.6226 + 0.7790]
  = 0.7008

On a unit-sphere Winkel Tripel map, Tokyo lands at approximately (x = 1.79, y = 0.70). Multiplying by the radius R of the sphere-as-mapped — for a map 360 pixels wide on a 1-radian-unit sphere, R = 360/π/2 ≈ 57.3 — gives Tokyo at about (102.5, 40.1) pixels east-of-prime, north-of-equator on the map.

The arithmetic-mean construction means that Tokyo's final position is exactly halfway between where the equirectangular projection would place it and where the Aitoff projection would place it. This averaging dampens the worst distortions of either input projection.

Usage in publications

Beyond National Geographic, Winkel Tripel appears in:

• The CIA World Factbook uses Winkel Tripel for its world political map and for many of its country-locator maps.
• The United Nations uses Winkel Tripel in some of its statistical yearbook materials.
• Time Magazine and several other US magazines adopted Winkel Tripel for their world-map graphics following National Geographic's switch.
• Microsoft Encarta used Winkel Tripel in its world atlas software before the encyclopedia was discontinued.
• xkcd's “What Your Favorite Map Projection Says About You” comic strip (xkcd 977, 2011) listed Winkel Tripel as the choice of “a baby boomer who likes solid, dependable maps” — a half-joking but accurate cultural placement.

The projection's popularity is driven less by its mathematical properties than by its visual legibility and the National Geographic endorsement. For thematic data that requires equal-area treatment — choropleth maps showing population density or economic indicators — cartographers use equal-area projections covered in the /learn/equal-area-projections support. For general-purpose reference maps, Winkel Tripel remains the default.

Software support

Winkel Tripel is well supported in modern cartographic software:

• PROJ: +proj=wintri with optional lat_1 parameter
• D3.js: d3.geoWinkel3() from the d3-geo-projection package
• GDAL / QGIS: built-in as a standard projection
• MapLibre / Mapbox: available as a runtime projection
• Python (pyproj, cartopy): standard
• R (rnaturalearth, sf, ggplot2): standard

The closed-form formula simplifies the implementation considerably compared to Robinson — there is no lookup table to ship, just an equation. This makes Winkel Tripel marginally cheaper to compute and to serialise.

For tile-based zoom-and-pan web maps, Winkel Tripel is rarely used because the tile-pyramid system used by Google Maps, OpenStreetMap, and most web-mapping software is built around Web Mercator (covered in the Mercator support). Winkel Tripel remains the preferred choice for static thematic world maps where the entire globe is shown at once and where the viewer is not expected to zoom in to street-level detail; for the latter case, Web Mercator's tile-pyramid efficiency continues to dominate.

Sources

• Snyder, Flattening the Earth (University of Chicago Press, 1993) — the definitive history including Winkel and his contemporaries.
• Snyder, Map Projections — A Working Manual (USGS Prof. Paper 1395) — formulas and projection mathematics.
• Robinson, Snyder, Voxland, Album of Map Projections (USGS Bulletin 1453, 1989) — comparative reference to many projection families.
• National Geographic Society — projection adoption history.

For closely related topics, see /learn/robinson-projection for the projection Winkel Tripel replaced at National Geographic, /learn/peters-projection for the controversy that Robinson and Winkel Tripel both sidestepped, and /learn/equal-area-projections for the alternative family that preserves area exactly.