Cartographic Projection
every map is wrong, and by exactly how much
Theme

The thread: The trade-off is forced

Conformal means the principal scales are equal; equal-area means their product is one. Both at once forces them both to one, which is an isometry, which cannot exist. The trade-off is two lines of algebra.
0°20°40°60°80°MercatorGall–PetersWinkelRobinsonlatitudeangular deformationalong a meridian What each projection optimises

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

 equator23°45°60°70°1×MercatorEquirectangularMillerGall–Peterslatitude of the cell1× means treated fairlyrelative to the equator What is taught wrongly

Mercator against Peters

The most-argued question in cartography, conducted almost entirely without anyone measuring anything. Both projections are exactly what they claim, each destroys what the other keeps, and the numbers are computable in either direction.

 EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree 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.

 0°20°40°60°80°exact at equatorexact at 30°exact at 45°latitudeangular deformationall three are equal-area The families

What a standard parallel buys

A standard parallel is a line where the projection is exact. Choosing one does not reduce the distortion — it decides where the distortion is zero and lets everything grow away from it.

 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured The impossibility

The trade-off is two lines

Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.

 0°20°40°60°80°MercatorGall–PetersMollweideWinkellatitudeangular deformationalong a meridian Measuring distortion

The two ways a map is wrong

Angles and areas fail independently. A projection can be perfect about one and catastrophic about the other, and a single number summarising "how distorted" a map is has already thrown away the distinction that matters.

 RobinsonWinkel tripelMollweideMercatorGall–PetersEckert IVsame sphere, same graticuleno two agree What each projection optimises

Compromise projections

A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.

 Mercator5.62× areaGall–Peters1.00× areaMollweide1.00× areaEquirectangular2.36× areathe cell at 60°scaled to fit, areas as stated What is taught wrongly

The projection that shows true size

There is no such thing, and the phrase hides a real question. Equal-area projections preserve area and destroy shape; nothing preserves size in the sense the phrase implies; and the tools that make the point best are not maps at all.

 10⁻⁶10⁻⁴10⁻²110010⁻¹¹10⁻⁸10⁻⁵10⁻²1010000maximum angular deformation / degreesmaximum areal errornothing here, everMercatorStereographicWeb MercatorGall–PetersMollweideEquirectangulartolerances: 0.0001° and 0.00000121 projections measured Measuring distortion

Measuring instead of naming

A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.

All themes