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

The same point at 30°, 55° under four projections

Drawn here at the parameters it defaults to, with every essay that calls it.
The same point at 30°, 55° under four projections. One circle on the sphere, four projections, four ellipses. A conformal projection keeps it circular and lets the area run; an equal-area projection keeps the area and lets the shape go. Nothing keeps both, and the dashed circle shows what keeping both would look like.

One circle on the sphere, four projections, four ellipses. A conformal projection keeps it circular and lets the area run; an equal-area projection keeps the area and lets the shape go. Nothing keeps both, and the dashed circle shows what keeping both would look like.

It is drawn by indicatrix-figure with show: "indicatrix-compare" — one member of a family of 8 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

7 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

The indicatrix at 45°, 55° on Mercator. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.743 and b = 1.743; their product is the areal factor 3.040; and the maximum angular deformation is 0.00°. h and k are shown too, and depend on the coordinates rather than on the map. Measuring distortion

Tissot's indicatrix

A tiny circle on the sphere becomes an ellipse on the map, and the ellipse's two axes are the whole story. Almost every published indicatrix is drawn without them, which discards the content and keeps the decoration.

 Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the four named here are Mercator, Mercator (ellipsoidal), Web Mercator, Gall–Peters. What is taught wrongly

Web Mercator is not conformal

It carries almost every map on the internet, it is named after the projection whose entire purpose is preserving angles, and it does not preserve angles. The machinery here found that without being told to look.

 The indicatrix at 30°, 40° on Gall–Peters. A circle on the sphere becomes this ellipse on the map. Its semi-axes are the two principal scale factors, a = 1.083 and b = 0.923; their product is the areal factor 1.000; and the maximum angular deformation is 9.16°. h and k are shown too, and depend on the coordinates rather than on the map. Measuring distortion

What survives a change of coordinates

The scale along the meridian is a property of the map and the grid together. The principal scale factors are a property of the map alone. Only the second kind describes the projection, and the two are routinely quoted as though they were the same thing.

 How much each projection inflates a cell, by latitude. Five patches of the sphere, each 20° by 10°, and the factor by which each projection enlarges them relative to the equatorial one. An equal-area projection sits flat on 1. Mercator reaches 15.4× at 70°, which is the mechanism behind every complaint about the size of Greenland. 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.

 Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the five named here are Mercator, Stereographic, Gall–Peters, Mollweide, Winkel tripel. 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.

 How large a map has to be before a wrong projection stops fitting it. Three rivals fitted to a Mercator graticule centred at 45° north, over regions from 1° to 70° of half-extent. The residual is the shape difference alone, with the best scale, rotation and offset removed, and both axes are logarithmic. The horizontal rule is a fifth of a per cent of the map's width — about the width of a drawn line on a printed sheet — and where a curve is below it, no measurement of that map can tell the two projections apart, however carefully it is made. What is taught wrongly

Two projections that cannot be told apart

Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.

 The circle a reprojection is about to turn into an ellipse. A circle drawn on the Mollweide plane, and its image under the map that carries that plane to Mercator — normalised so each pair has the same area, because the whole map can be rescaled and the shape is what is being shown. The dashed circle is what an undistorted reprojection would leave. This is Tissot's construction with a plane in place of the sphere, and it is the right picture for a reprojection because both ends are pictures. Worst angular deformation over the sampled points: 81.2°. Drawn on the source plane, in Mollweide. Measuring distortion

A projection between two projections

Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.

The whole library