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

The area of one 20° × 10° cell at 50–60° north, seven ways

Drawn here at the parameters it defaults to, with every essay that calls it.
The area of one 20° × 10° cell at 50–60° north, seven ways. The cell has an exact area — R²Δλ(sin φ₂ − sin φ₁), 1,416,580 square kilometres — so every other row is a measurement of the method rather than of the ground. The equal-area projection returns it to 1.000000 and the spherical polygon formula to 1.000000, which is three routes agreeing — and the same cell integrated on the ELLIPSOID comes out 0.45 per cent away from all three, because the sphere is a model. Taking the shoelace in Mercator gives 3.06 times too much, and treating degrees as a length gives 1.75 times — about sec φ at the cell's middle, which is where that error comes from.

The cell has an exact area — R²Δλ(sin φ₂ − sin φ₁), 1,416,580 square kilometres — so every other row is a measurement of the method rather than of the ground. The equal-area projection returns it to 1.000000 and the spherical polygon formula to 1.000000, which is three routes agreeing — and the same cell integrated on the ELLIPSOID comes out 0.45 per cent away from all three, because the sphere is a model. Taking the shoelace in Mercator gives 3.06 times too much, and treating degrees as a length gives 1.75 times — about sec φ at the cell's middle, which is where that error comes from.

It is drawn by operation-figure with show: "area-methods" — one member of a family of 14 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.

8 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.

"Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west. What a machine does with it

A degree is not a unit of length

"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.

 The area of one 20° × 10° cell at 50–60° north, seven ways. The cell has an exact area — R²Δλ(sin φ₂ − sin φ₁), 1,416,580 square kilometres — so every other row is a measurement of the method rather than of the ground. The equal-area projection returns it to 1.000000 and the spherical polygon formula to 1.000000, which is three routes agreeing — and the same cell integrated on the ELLIPSOID comes out 0.45 per cent away from all three, because the sphere is a model. Taking the shoelace in Mercator gives 3.06 times too much, and treating degrees as a length gives 1.75 times — about sec φ at the cell's middle, which is where that error comes from. What a machine does with it

Computing an area needs a surface

A shoelace over a ring of coordinates returns a number whatever the coordinates are. For one twenty-by-ten-degree cell it returns 1.75 times the true area in degrees, 3.06 in a conformal plane, and exactly the closed form in an equal-area one — and the closed form itself is 0.45 per cent out, because the sphere is a model too.

 A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres. What a machine does with it

The antimeridian is a cut in the numbers

A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.

 five planes a dataset might be stored in, scored on three operations. Each candidate measured over -10° to 30° east and 35° to 60° north: the worst areal error, the worst angular deformation, and the spread of the scale factor, which are what an area query, a shape and a distance respectively depend on. The best plane for areas is Gall–Peters, for shapes Lambert conformal conic, and for distances Lambert conformal conic — three different answers, and no fourth candidate would collapse them, because a projection exact in two of these columns has a = b = 1 everywhere and is the isometry Gauss's theorem forbids. area of a polygon costs 3.06× too large in the wrong plane; drawing a line between two points costs 194 km from the ground it claims. What a machine does with it

The operation decides the coordinate system

Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.

 Where the middle of a 20° × 20° region is, in five planes. The region is drawn in longitude and latitude — which is itself a projection, and one of the ones being compared. Each filled mark is the shoelace centroid computed in one projected plane and inverted back to the ground; the hollow mark is the centre of area on the sphere, by integration. They spread over 273 kilometres. The equal-area member is 66 kilometres out, because a centroid is a first moment and preserving area says nothing about where the area sits. What a machine does with it

A centroid belongs to a plane

Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.

 The picture is kept, at four tolerances. One closed curve of 3001 vertices, simplified at four tolerances. Douglas–Peucker's promise holds in every panel: no discarded vertex is further than ε from the line drawn in its place, measured at 0.1158 against 0.128 in the last. The picture survives. The enclosed area does not: it falls by 5.43 per cent, and it falls rather than wandering, because cutting a corner takes area off and never puts it back. What a machine does with it

A tolerance is a promise about the picture

Douglas–Peucker guarantees exactly one thing: no vertex it discarded is further than ε from the line drawn in its place. It says nothing about the enclosed area, nothing about which side of the boundary a point ends up on, and nothing about whether the curve still fails to cross itself — and all three are what the geometry is usually being asked.

 A ring round the pole at 80°, and the two pieces it makes. the boundary of a small polar cap — and of everything else. A closed curve divides a sphere into two pieces and neither of them is the outside: one is 4 thousand square kilometres and the other is 506 thousand, a ratio of 130.6 to one, and the coordinates are the same either way. The two colours are the two pieces, sampled at points rather than shaded, because shading one of them would already be the decision this figure is about. What a machine does with it

A polygon on a sphere has no outside

Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.

 What simplifying a boundary does to the number stored beside it. One region, simplified at five tolerances, with the error in the two quantities a consumer computes from the pair. If the density was stored, the total it implies moves by exactly the area's error — -1.55 per cent at the loosest tolerance. If the total was stored, the density it implies moves the other way by the same amount. Nothing in the file says which of the two was measured and which is being derived, and the simplification is normally done by a tool that never opens the attribute table. What a machine does with it

The attribute is a claim about the geometry

Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.

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