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

Shoelace — where it appears

The planar formula for a polygon's area from its vertices, which returns a number whatever coordinate system those vertices happen to be in. It returns a plane's area, so applying it to longitudes and latitudes measures a region on the plate carrée rather than on the ground.

Named by 5 essays across one field — each of them below, with the objects they name alongside 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.

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.

applied · Dataset 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.

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.

applied · Dataset 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.

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.

applied · Dataset 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.

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.

applied · Dataset A parent and its children, twice. An aperture-7 hexagonal hierarchy beside a square one. The heavy outline is the parent and the light ones are its children. On the right every child is wholly inside and the four of them tile the parent exactly. On the left the child lattice is turned by 19.107° relative to the parent's, only the central child is wholly inside, and 7.14% of the parent is covered by no child of its own. The two families have exactly the same total area — a hexagon cannot be tiled by smaller hexagons at any ratio at all, which is why the mismatch is a construction rather than an approximation.

A cell's children do not fit inside it

Ten rungs price one cell system at one resolution, and every one of them is used hierarchically. A hexagonal hierarchy does not nest: at the aperture-seven scheme the discrete global grids use, one fourteenth of a parent is covered by no child of its own, exactly, and each of the six ring children is eleven twelfths inside.

applied · Cells

Named alongside it

The objects these essays reach for when they reach for this one.

Closed formConventionVerificationAntimeridianAreaAreal scaleBoundaryEqual-areaNumerical integrationToleranceTopologyAggregation

All concepts