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
Asked across the counter
What is a rhumb line?
The track you follow by holding one compass bearing the whole way — a loxodrome. In 1537 Pedro Nunes proved it is not the shortest route: except along a meridian or the equator it is a spiral that circles the pole forever without arriving. The shortest route is the great circle.
Why is every map a lie?
Because 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. Tissot proved in 1859 that none can keep both area and angle anywhere but at isolated points, which is why every row of the table above has a “no” in it.
What is a Tissot indicatrix?
A small circle drawn on the globe, then shown as the projection actually renders it. Round means the shape survived at that point. Large means the area did not. A field of them across a sheet shows you exactly where the projection is honest.
How is the distortion measured?
Take the derivative of the projection east and north at a point, with the
eastward one divided by cos φ. The two singular values
σ₁ ≥ σ₂ of that 2×2 matrix are the scale factors along the
principal directions, and everything follows:
areal scale = σ₁·σ₂
angular deformation = 2·asin((σ₁−σ₂)/(σ₁+σ₂))Area is kept where the first is 1; angle is kept where the second is 0.
Is Mercator a bad map?
Mercator is an excellent instrument that happens to be a bad world map. Built in 1569 to draw a constant-bearing course as a straight line — exactly what a navigator needs, and it does it perfectly. It is a poor wall map because the areal cost at high latitude is severe: Africa is about fourteen Greenlands, and on Mercator they look comparable.
Which projection should my chart use?
It depends entirely on what the sheet is for. Plotting a course by compass — Mercator, because the rhumb is straight. Comparing the size of countries — an equal-area projection. Measuring distance from one place — azimuthal equidistant, centred on it. We are asked for Mercator and we ask what the sheet is for; about half the time the answer changes the projection.
How do I commission a sheet?
Write to studio@webstoree.com or call +91 89815 64862. Say what the sheet is for, which waters it covers, and at what scale it will be read. That is enough to settle the projection before anything is drawn.