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

Taxonomy — where it appears

A classification of projections into families, usually by the developable surface a construction would have used. It is a description of the output rather than of a construction, which is why a projection can be called cylindrical without ever having been on a cylinder.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Construction against property. Every projection in the library, sorted by the shape it was notionally rolled from and by what it actually preserves. The families do not line up with the properties: cylindrical and pseudocylindrical and pseudoazimuthal and azimuthal and conic each contain projections of more than one kind, which is why the cylindrical–conic–azimuthal taxonomy answers a question nobody has. Three entries are picked out: Mercator, Albers equal-area conic, Orthographic.

Cylinders, cones and planes

The standard taxonomy sorts projections by the shape of paper they were notionally rolled from. It is memorable, it is how everyone is taught, and it says almost nothing about the properties anyone actually chooses on.

families · Families American polyconic. The graticule of the American polyconic projection at 30° of longitude and 15° of latitude. a different tangent cone for every parallel — true to scale along all of them, and along the central meridian. It is neither conformal nor equal-area.

The projections that gave up being one thing

The polyconic is built from a different cone for every parallel, which means it is built from no cone at all. It preserves nothing the usual tests look for, it has an exact property neither of them measures, and its sheets do not fit together — a defect discovered in the field rather than at the drawing board.

families · Families The equal-area conic, from cylinder to plane. The largest distance between the conic at cone constant n and each of its two limits, over a shared grid, with the free scale and offset removed. Both fall as the FIRST power of the distance from their end — the slope on these axes is one — so the conic is never nearly cylindrical: halving n only halves the difference. At n = 0.999999 the conic is the Lambert azimuthal to 1.3e-6, and at n = 0.000001 it is the Lambert cylindrical to 1.2e-6.

The conic is the whole family

Cylindrical and azimuthal are usually presented as two of three families beside the conic. They are the two ends of it. One parameter runs from the cylinder to the plane, and both limits are exact rather than suggestive — which is measurable, and measured here.

families · Families The azimuthal family, as one function of one variable. Every azimuthal projection is a rule for how far from the centre to draw a point at angular distance ρ, and nothing else. five named projections are plotted as their radial functions — tan ρ, 2 tan(ρ/2), ρ, 2 sin(ρ/2), sin ρ — and the two dashed curves are the solutions of the two conditions, integrated from the centre with no more than "true scale at the centre" to start from. They land on the stereographic and the Lambert azimuthal to 8e-10. The names are the answers to the equations, not descriptions of viewpoints.

The azimuthal family is one function

Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.

families · Families What is left after the best rigid motion of the page. Every projection in the library, rotated about its own symmetry axis by five degrees, with the best rigid motion of the page fitted and the leftover measured against the picture's own size — on a logarithmic scale, because the answers span four orders of magnitude. Fourteen sit at 2e-6 or below, which is the axis search's own floor. Seven sit between 9e-3 and 3e-2. There is nothing in between, so the split is a fact rather than a threshold — and the fourteen are exactly the cylinders, the cones and the planes.

The family is a symmetry, not a shape

Rung seven dissolves the taxonomy: eleven named maps fitted to one perspective formula, six reproduced exactly and five refused, so the developable surface describes how the projections were discovered rather than what they are. What is left is the question it does not ask. If the shapes are not the classification, what is?

families · Families Two members, their average, and the family's best answer to it. Two equal-area conics at cone constants 0.25 and 0.85, the average of the two, and the member of the family nearest that average — at 0.540, which is not the parameter midpoint. The average is not a conic at all: its parallels are still arcs but they are arcs of circles about different centres, so no single cone constant reproduces it and the residual is 0.1722.

A family is not closed under averaging

Nine rungs treat a family as a set of maps with a parameter running through it, and this collection's own compromise projections are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.

families · Families

Named alongside it

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

AzimuthalConeConicDevelopable surfaceCone constantCylinderCylindricalOne-parameter familyAspectEqual-areaInvariantOptimisation

All concepts