What each projection optimises

Which projection is best

An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.

It is the first question anybody asks and it has no answer, which sounds evasive and is the most useful thing in the subject.

6 projections of the same sphereThe same graticule under equirectangular, mercator, mollweide, sinusoidal, robinson, winkelTripel. Every one of them is a faithful drawing of the same object and no two agree, because each has chosen a different thing to preserve.EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree
Fig. 1 Six projections of the same sphere. Every one is a correct drawing of the same object, and no two agree, because each has chosen a different thing to keep. Asking which is best is asking which choice is correct without saying what the choice is for.

Why the question is incomplete

No projection preserves both angles and areas, and most preserve neither exactly. So every projection is a decision about what to sacrifice, and a decision cannot be evaluated without knowing what it was in aid of.

That is not a philosophical dodge. It has an operational form: name the property the map must have, and the question usually answers itself.

  • A map for compass navigation must render constant bearings as straight lines. That is Mercator, uniquely and exactly.
  • A map for plotting shortest routes must render great circles as straight lines. That is the gnomonic projection, and it is the only one.
  • A map showing a quantity per unit area — population density, land use, disease incidence — must not distort area, or the visual impression contradicts the data. Any equal-area projection.
  • A map for measuring angles or preserving local shape at large scale — a survey, a topographic sheet — must be conformal. Lambert conformal conic or transverse Mercator, depending on the region’s shape.
  • A map for looking at the world in general has no exact requirement, which is what compromise projections are for.

Five purposes, five different answers, and no contest between them.

What settles it in practice

Three questions, in order.

What must be true of the map? The property. If there is a hard requirement, it eliminates almost everything.

What is the extent? The distortion a map must absorb is bounded below by the curvature it covers, so a small region admits almost any projection and a world map admits none comfortably. For a city, the choice barely matters and any conformal projection at a suitable scale is fine. For a hemisphere it matters a great deal.

What is the shape of the region? This decides the family and the aspect more than anything else does. A region long in longitude and narrow in latitude — Russia, the United States — suits a conic with two standard parallels. A region long in latitude and narrow in longitude — Chile, Norway — suits a transverse cylindrical projection. A roughly circular region suits an azimuthal one centred on it.

That last question is the one most often skipped, and it is why the aspect is a free choice worth exercising.

What a standard parallel buysThree equal-area cylindrical projections differing only in where they are exact. Each has zero angular deformation at its own standard parallel and grows away from it in both directions. Choosing a standard parallel is choosing which latitudes to treat well, and there is no choice that treats them all well.20°40°60°80°exact at equatorexact at 30°exact at 45°latitudeangular deformationall three are equal-area
Fig. 2 Three equal-area projections differing only in where they are exact. All three preserve area perfectly; each has zero angular deformation only at its own standard parallel. Even after the family is chosen, there is a choice left, and it decides which latitudes are treated well.

The distortion goes somewhere

A useful way to think about the choice: a projection cannot reduce the total distortion, only redistribute it.

Choosing a standard parallel puts the zero there and pushes the error away from it. Choosing an aspect rotates the whole distortion pattern to fall where it does least harm. Interrupting the map cuts it into pieces so each carries less — at the cost of tearing the map apart.

None of these makes the distortion smaller in aggregate. They decide where it lands, and a well-chosen projection is one whose distortion lands where nobody is looking.

The answers that are actually wrong

Very little in this field is simply wrong, and a few things are.

Mercator for a general world map. Not because Mercator is bad — it is exactly what it claims — but because the areal distortion is fifteenfold at 70° and nothing about a general-purpose map needs constant bearings. This is the substance of the Peters argument, and the substance is right.

Any rectangular projection for a world map. Seven North American geographic organisations issued a joint resolution to this effect in 1989, against Mercator, Gall–Peters and the rest alike. The objection is that a rectangular world map implies the poles are lines rather than points, which is a topological misstatement rather than a matter of degree.

A conformal projection for a thematic map of area-based data. A choropleth on Mercator makes high-latitude countries look important in proportion to their inflation. The data is right and the impression contradicts it.

Web Mercator for anything measuring the ground, since it is not conformal and much software assumes it is.

Construction against propertyEvery projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has.conformalequal-areacompromiseclaim failscylindricalMercatorMercatorLambertGall–PetersBehrmannEquirectangularMillerWebpseudocylindricalSinusoidalMollweideEckertRobinsonpseudoazimuthalHammerWinkelazimuthalStereographicLambertOrthographicGnomonicAzimuthalconicLambertAlbersrows: how it is builtcolumns: what it preserves
Fig. 3 Every projection in the library, sorted by construction and by property. The families do not line up with the properties — several construction families contain projections of more than one kind, which is why choosing by family answers a question nobody has.

That table is the argument against the usual teaching order. Being told a projection is cylindrical says how it was built and almost nothing about what it preserves, and the property is what the choice depends on.

Compromise as a legitimate answer

For a general-purpose world map the honest answer is that no property is required, and a projection that preserves nothing exactly may distort everything less than one that preserves something exactly.

Angular deformation against latitude, four projectionsThe same quantity for mercator, gallPeters, winkelTripel, robinson, from the equator to 80°. A conformal projection sits flat on zero in the first of these plots and runs away in the second; an equal-area projection does the reverse. Nothing is flat in both.20°40°60°80°MercatorGall–PetersWinkelRobinsonlatitudeangular deformationalong a meridian
Fig. 4 Angular deformation against latitude. Mercator is flat on zero and pays elsewhere. The equal-area projections rise steeply. Winkel tripel and Robinson sit between, worse than Mercator on angle and far better than Gall–Peters, while being far better than Mercator on area.

That is the case for compromise projections, and it is why National Geographic moved from Robinson to Winkel tripel rather than to anything exact. A world map’s job is to look like the world, which is not a property with a formula.

The question behind the question

People asking which projection is best are usually asking something else: which map belongs on the wall, or which is the fair one.

Neither has a geometric answer. Fairness is not a property a projection can have, because fairness is about what a map is used to say and a projection is a function. What can be said is which distortions a given choice imposes and how large they are, which is measurable, and this site measures it.

Past that point the decision is editorial, and it should be made by someone who knows it is editorial.

Two questions that settle most cases

Past the property and the region, two further questions resolve nearly everything left.

How large is the region relative to the sphere? The distortion forced is proportional to the curvature carried, so this fixes how much the choice matters before it fixes what to choose. Below about a degree across, any reasonable projection is fine and the choice is administrative. Above a hemisphere, nothing is fine and the choice is a matter of which compromise to accept.

Where is the region relative to the projection’s axis? The aspect is free and rotating the projection so its zero-distortion line runs through the region is the cheapest available improvement. This is the question most often skipped, and skipping it means accepting a projection optimised for the equator regardless of where the map is.

Lambert conformal conicThe graticule of the Lambert conformal conic projection at 30° of longitude and 15° of latitude. conformal, and what aeronautical charts are drawn on. It is conformal.conformaldrawn in Lambert conformal conic
Fig. 5 The Lambert conformal conic with standard parallels at 20° and 60°. Conformal everywhere, exact along both parallels, and shaped to suit a region wide in longitude and shallow in latitude — which describes most mid-latitude countries.

What software defaults do

Worth knowing, because in practice most projections are chosen by not choosing.

A web mapping library defaults to Web Mercator. A desktop GIS defaults to whatever the first loaded layer used. A statistical plotting package defaults to equirectangular, because that is what treating longitude and latitude as Cartesian coordinates produces.

That last case is the most common and the least deliberate. A scatter of points plotted with longitude on the x axis and latitude on the y axis is a plate carrée, and it is a projection with 108° of angular deformation and eightfold areal error at high latitudes. Nobody chose it; it is what happens when the coordinates are treated as plain numbers.

How Equirectangular distorts, by latitudeAngular deformation and areal scale plotted against latitude along the meridian at 0°. On Equirectangular the angular deformation reaches 114.2° and the areal factor reaches 11.5.-60°-30°30°60°angular deformation, to 114°areal factor, to 11.5×latitudetwo independent distortionsalong the 0° meridian
Fig. 6 The plate carrée, which is what plotting latitude against longitude produces. Both distortions grow steadily with latitude, and this is the projection most data visualisations use without anyone deciding to.

Scale changes the answer

One more axis, and it cuts across everything above.

At street scale the choice barely matters and consistency matters more — which is why national grids exist and why using one is nearly always right for local work regardless of its projection’s properties.

At country scale the choice matters and the region’s shape decides it.

At world scale nothing works and the honest answer is a compromise, or a globe, or an interaction that sidesteps the problem.

The mistake worth avoiding is carrying a world-scale intuition to a local problem. The Mercator arguments that matter for a wall map are irrelevant for a city map, where the areal distortion across the extent of the city is a fraction of a per cent.

A worked example

Following the questions through on one case makes the method concrete.

The map: a thematic map of population density for Chile.

What must be true? Density is per unit area, so areal distortion would contradict the data. Equal-area is required.

How large? Chile spans about 38° of latitude and 5° of longitude. That is a substantial extent in one direction and a narrow one in the other.

What shape? Long north–south. So the projection’s zero-distortion line should run north–south, which means a transverse aspect.

Answer: a transverse cylindrical equal-area projection, or a Lambert azimuthal equal-area centred on the country. Not a normal-aspect anything, and not a conic — a conic’s standard parallels run east–west, which is the wrong way round for Chile.

Four questions, no contest, and the taxonomy never came up.

Lambert azimuthal equal-areaThe graticule of the Lambert azimuthal equal-area projection at 30° of longitude and 15° of latitude. equal-area, and the standard choice for a hemisphere. It is equal-area.equal-areadrawn in Lambert azimuthal equal-area
Fig. 7 The Lambert azimuthal equal-area projection. Equal-area everywhere — measured at 1.5×10⁻⁹ — and symmetric about a centre that can be put wherever the map is, which makes it the general-purpose answer for a compact region.

The question that has no geometric content

One more framing worth having, because it clears away most of the arguing.

“Which projection is best” is really two questions wearing one coat. Which projection best serves this purpose is answerable and this essay answers it. Which projection should the world use by default is a question about defaults, audiences and institutions, and geometry contributes nothing to it beyond the constraints.

The second question is legitimate and it is not a cartographic question. It was argued as one for twenty years, which is why the argument did not converge.

Tissot's indicatrix across RobinsonA small circle on the sphere, drawn where the projection puts it. The dashed circle behind each is what an undistorted map would show. On Robinson ω reaches 49°, and the areal factor reaches 1.2.dashed: an undistorted circledrawn in Robinson
Fig. 8 Robinson’s indicatrices. No clean pattern, no exact property, moderate distortion everywhere — which is what a projection chosen for a purpose with no hard requirement looks like when measured.

The shortest useful answer

If a single recommendation is wanted, here is one that is defensible more often than not.

For a world map: Winkel tripel or Robinson. No exact property is required and both distort everything moderately.

For a thematic map of anything per unit area: any equal-area projection, chosen by region shape.

For a region smaller than a continent: a conformal projection with standard parallels fitted to the region, or simply the national grid, which is what everything else in that country will be in.

For navigation or bearings: Mercator.

For shortest routes: gnomonic.

Every one of those is a purpose paired with a property, which is the whole method compressed into five lines.

A final caution about all of the above. Every recommendation here assumes the map is being made rather than inherited, and most maps are inherited — the projection arrives with the data, the basemap or the organisation, and changing it costs more than the improvement is worth. Knowing what the inherited projection does is then more valuable than knowing what a better one would have been.

The question also has a good answer that is rarely given: ask what the map is for, and if the answer is “general purposes” then no property is required and the choice is editorial. Most people asking which projection is best are making a general-purpose map, which means the honest answer is that the geometry does not decide it — and saying so is more useful than naming a projection.

One more thing the question hides: it assumes a single map. A great deal of good cartographic practice is to use several — a conformal one for shapes, an equal-area one for quantities, an azimuthal one centred where the subject is. The projections are cheap and the assumption that one map must serve every purpose is the constraint doing the damage.

What was computed here

Every projection drawn is measured rather than described: its angular deformation and areal error are computed from its own derivatives at several hundred points, and the claims are asserted before any caption is written.

The family table is generated from those measurements rather than from a taxonomy, and it carries an assertion: at least two construction families must contain projections of more than one property class. If the families ever lined up neatly with the properties, the figure’s argument would be false and the build would stop.

The standard-parallel figure asserts that each variant’s distortion minimum falls at its own standard parallel, to within two degrees — which is the check that the projections are what they say and the plot is reading them correctly.

What the pictures cannot show

The suitability of a projection for a purpose, which is the only thing this essay is actually about. The figures show distortions; whether a given distortion matters depends on the map’s use, its audience and its scale, none of which is in the geometry.

The grid figure also flatters the compromise projections slightly, because at thumbnail size the differences that matter most — the fine behaviour near the poles and at the edges — are below the resolution of the drawing.

Who found it, and when

The recognition that projection choice is a design decision rather than a search for the correct answer is relatively modern as a stated principle, though it was obvious in practice to anyone making maps for a living.

John Snyder’s Map Projections: A Working Manual (1987) is the standard reference and is organised around exactly this: purposes first, projections second. Snyder was a chemical engineer who took up projections as a hobby and ended up at the US Geological Survey, and he designed the Space Oblique Mercator for satellite imagery — a projection built to a specification so narrow that it is useless for anything else and perfect for that.

Which is the principle in its purest form.

Where this goes next

What each projection is optimising is every projection minimises something. The projections that optimise nothing exactly are compromise projections. And the theorem that makes the choice unavoidable is the trade-off is two lines.