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

Branch cut — where it appears

The point or line at which an otherwise analytic map stops being analytic, which on a polyhedral face map is a corner and cannot be removed. It is where a polyhedral map's curvature is concentrated, and it is the one point of such a map at which no indicatrix exists because no linear approximation does.

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

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 The conformal map onto a square face of a cube. The spherical face of a cube carried onto its flat face by a map that is conformal everywhere — the measured angular deformation over the drawn interior is 1.63e-6°, which is the arithmetic's own floor. The rings and spokes are circles and radii on the sphere, and they cross at right angles here because that is what conformal means. The map was solved for as 16 terms of a series rather than written down: the face's edge comes out straight to 0.33 per cent of its own half-width, and that residual — not the conformality — is what more terms buy. At each corner the map behaves like ζ^0.75, so the scale factor there is infinite.

A conformal map onto a face

The polyhedral ladder ended owing a conformal face map, on the grounds that it needs elliptic functions. It does not: a conformal map of the sphere is an analytic function of one conformal coordinate, so the map is a power series, choosing it is a least-squares fit — and its scale factor is infinite at the corners, which is the angle deficit arriving as a singularity.

families · Polyhedral What a solved map is, as a list of numbers. A map with no formula is a list of coefficients, and this is the list. The Chebyshev map of an elongated region, 30° by 10° has its coefficients falling by a factor of 1.5e+13 from the first to the fourteenth, so a table of a dozen numbers carries the whole projection; the conformal cube face's coefficients, marked separately, fall far more slowly because the map has a singularity at each corner. How fast this line falls is exactly how portable the map is — and neither map has a name, an inverse in closed form, or a formula anybody could quote.

A map with no formula

The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.

choosing · Condition

Named alongside it

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

Closed formConvergence rateLeast-squaresSeries truncationAngle deficitAntimeridianBoundaryBounding boxConformalityConventionCoordinate reference systemEPSG

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