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

Duality — where it appears

The exchange of faces for vertices in a polyhedron, which turns a triangulation to draw on into a tiling to count in. It is why a geodesic grid's cells are hexagons: they are the dual faces of a subdivided icosahedron's vertices, and twelve of them must be pentagons.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Which edges to cut is a spanning tree. The icosahedron's faces as nodes and its 30 shared edges as links. A net keeps 19 of those joins and cuts the rest, and the joins have to form a spanning tree — connected, so the net is one piece, and acyclic, so it lies flat. The heavy links are one such tree. The number of distinct nets is therefore the number of spanning trees of this graph, which Kirchhoff's theorem gives as a determinant: 5,184,000 for the icosahedron.

The cut has to go somewhere

A solid lies flat only if it is cut open, and which edges to cut is a spanning tree of the face graph — so the icosahedron has exactly 5,184,000 distinct nets, a determinant rather than an estimate. All 384 of the cube's were laid flat and tested: not one overlaps, while an irregular tetrahedron overlaps in four of its sixteen.

families · Polyhedral Tissot's ellipse and the one that governs gradients, at 20°E 48°N. The solid ellipse is the image of a small circle — Tissot's indicatrix, semi-axes a and b. The dashed one is the image of a unit gradient, whose semi-axes are 1/b and 1/a because a gradient transforms by the inverse transpose of the Jacobian rather than by the Jacobian. Its long axis therefore lies where the indicatrix's short one does. On Mercator both are circles, so a gradient's direction survives; on Lambert cylindrical the two ellipses are the same shape turned through a right angle, so the worst direction for a gradient is the best direction for a shape.

A slope is not a shape

Every map in this collection has carried geometry. An applied map far more often carries a field — elevation, pressure, a density — and the first thing anybody does with one is differentiate it. A gradient is a covector, it transforms by the inverse transpose of the Jacobian, and the ellipse that governs it is the indicatrix turned inside out.

distortion · Gradient The same six contours on Mercator and Lambert cylindrical. The value of a harmonic sum with a summit and a basin in the northern mid-latitudes travels with the point, so the set of points at a stated level is the same set on every map and each contour is exactly right on both panels. Everything a reader measures from them is not: the spacing between neighbouring contours, their lengths, and the area between two of them all change from one panel to the other, and the two panels are the same field.

The contour is right and the reading is wrong

There is exactly one thing about a field that no projection can get wrong: which points share a value. The contour lines on any two maps of the same field are the same set of points. Every quantity a reader takes off them — the spacing, the length, the area between two of them, the hypsometric curve — is not, and the two kinds of map get different ones wrong.

distortion · Gradient Steepest descent on Lambert cylindrical, computed on the ground and on the page. Fifteen routes, each started at the same place twice. The solid line follows the true direction of steepest descent on the sphere; the dashed line follows the direction read off the page at every step, which is what an analysis of a projected grid does. Both take the same length of step on the ground, so the only difference between them is the direction. They part by up to 1517.8 kilometres, against a bearing error of 45.6°. The faint lines are contours of the field, which is seven caps at stated centres and widths.

Water runs downhill on the ground, not on the page

A drainage network is the set of steepest-descent trajectories of a field, so it is built entirely out of directions. A conformal map preserves those directions exactly and therefore preserves the whole network; an equal-area map does not, and sends a route up to 892 kilometres away from where the water actually goes.

distortion · Gradient Hilbert order on one face, as a curve. The order in which Hilbert numbering visits the 64 cells of one cube face at level 3. The line never leaves a cell without entering one that shares an edge with it — that is what makes it a space-filling curve, and it is why two cells with nearby identifiers are usually near each other on the ground.

The address is a curve through the sphere

A database does not fetch a set of cells, it reads ranges of identifiers — so the cost of a query is how many runs its cells form, not how many cells it needs. Hilbert order wins that measurement and loses the one usually quoted for it: its neighbouring cells are further apart in identifier than row-major's, on average and at worst.

applied · Cells

Named alongside it

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

ConformalityEqual-areaGradientTest fieldThematic mappingAnisotropyInvariantLocalityPrincipal directionScale factorTopologyAddress

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