Every map is a lie.
The sphere is curved and the paper is not, so something has to give — shape, or area, or distance, or direction. A projection is a decision about which. Turn the globe below and the page will measure what your decision cost, to three decimal places.
In 1537 Pedro Nunes, cosmographer to the Portuguese crown, proved that holding one compass bearing does not take you along the shortest route. It takes you along a spiral that circles the pole forever without arriving. Thirty-two years later Mercator built the projection that draws that spiral as a straight line, and every chart since has been an argument about what to trade for it.
Charts
Hydrographic sheets for the Atlantic ports, engraved and corrected against the notices every quarter.
Projection
We are asked for Mercator and we ask what the sheet is for. Half the time the answer changes the projection.
Magenta
The advisory colour on an Admiralty chart, and the only accent on this page. It marks what was measured, never what was chosen.
Turn the world
Drag the map. You are not panning an image — you are rotating the sphere underneath, and the projection is being evaluated again for every coastline point, sixty times a second. The ellipses are Tissot indicatrices: each one is a small circle drawn on the globe, shown as this projection actually renders it. Round means the shape survived. Large means the area did not.
How the distortion is measured
Take the derivative of the projection east and north at a point, with
the eastward one divided by cos φ because a degree of
longitude is a shorter distance near the pole. That is a 2×2 matrix.
Its two singular values σ₁ ≥ σ₂ are the scale factors
along the two principal directions, and everything follows:
areal scale = σ₁·σ₂
angular deformation = 2·asin((σ₁−σ₂)/(σ₁+σ₂))A projection keeps area where the first is 1 and keeps angle where the second is 0. Tissot proved in 1859 that no projection can do both anywhere but at isolated points, which is why every row of the table below has a “no” in it.
Africa is fourteen Greenlands
Below are the two outlines, drawn straight out of the projection with no rescaling — whatever size the projection makes them, that is the size you get. The areas underneath are computed twice: once on the sphere by spherical excess, once on the flat page by the shoelace formula, from the same coordinates the canvas is drawing.
The straight line that isn't
The rhumb line holds one compass bearing the whole way. The great circle is the shortest route there is. They are the same only along a meridian or the equator. On the left, Mercator — the rhumb is dead straight, which is the single thing the projection was built to do. On the right, the globe, where you can see what that straightness costs.
Mercator is not a bad world map. It is an excellent instrument, which happens to be a bad world map — and for four hundred years we have been hanging the instrument on the wall.
Nine ways to be wrong
All nine share the rotation you set on the drum. Every verdict beside them was measured in the browser a moment ago, not looked up — and not one of them keeps both.
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The hemispheres
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The Atlantic
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The bearing