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

Developable surface — where it appears

A surface that can be unrolled onto the plane without stretching, which is exactly a surface of zero Gaussian curvature — a cylinder, a cone or a plane. The sphere is not one, which is the whole of the impossibility, and most named projections are not projections onto one either.

Named by 11 essays across 2 fields — 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 A cylinder unrolls exactly; a sphere does not. Both pictures show the same grid. On the left it is wrapped round a cylinder of radius 1, on the right it is laid flat, and every distance in the grid is the same in both — the circumference is 2π and so is the width of the rectangle, checked to 10⁻⁹. This is possible because a cylinder has zero Gaussian curvature. No corresponding picture exists for a sphere.

What can be unrolled

A cylinder is obviously curved and is intrinsically flat, so it lays out on a table with nothing stretched. A sphere is not. The distinction is exactly zero Gaussian curvature, and it is why paper tubes exist and paper globes do not.

impossibility · Curvature 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 A direction carried once round the sphere. A vector transported round a closed loop on the sphere, kept as parallel to itself as the surface allows at every step — the component that leaves the tangent plane is removed and nothing else is done to it. The heavy arrows are its direction at the start and at the finish, drawn from the same point; the light ones are its direction along the way. It comes back turned through 282.0 degrees, which is 0.783 of a revolution, and nothing in the transport turned it. Drawn in an orthographic projection of the embedding, which is itself a map and has its own distortion.

A direction carried round a loop

Carry a bearing round a circuit, keeping it as parallel to itself as the surface allows, and it comes back turned. Round a parallel at 45° the turn is 4.443 radians and the cap enclosed is 1.840, and they sum to exactly one revolution — so the turning is not the curvature, and on a cone it is all of one and none of the other.

impossibility · Curvature How much curvature varies from place to place, body by body. The Gaussian curvature along a meridian, each body against its own smallest value, so the curves are comparable. Written from the bodies actually drawn: Earth varies by 1.0135, Mars varies by 1.0239, Vesta varies by 2.71, Phobos varies by 4.16, from the flattest place on each to the sharpest. A surface of constant curvature is a sphere and nothing else is, so the impossibility argument this field is built on has a local version on every real body: how flat a patch is depends on where the patch is.

Curvature that varies from place to place

The impossibility this site is built on was argued on a sphere, where the curvature is one number. On the Earth it varies by 1.35 per cent, on Jupiter by 31, on Vesta by a factor of 2.7 — and on a body with three axes it varies along a parallel, which no formula in latitude can express.

impossibility · Curvature Three surfaces with the same curvature, one of which is a sphere. Every one of these is a surface of revolution whose Gaussian curvature is 1 at every point, built by solving r″ + r = 0 for the meridian rather than by writing a shape down. The spindle closes to a point with an angle deficit, the sphere closes smoothly, and the bulge does not close at all — it ends in two circular edges. A surveyor confined to a patch of any of them, measuring angles and distances, cannot tell which one it is.

Two surfaces with the same curvature

The ladder's base says curvature is the obstruction to a faithful map. It has never asked whether curvature is the whole obstruction — and it is, locally: there is a whole family of surfaces with the unit sphere's curvature at every point, none of them a sphere, and a geodesic circle drawn on one agrees with the same circle on another to one part in 10¹⁴.

impossibility · Curvature One length, three corrugations. A straight segment shortened by a factor of 0.9, then wiggled across its own direction until its length is back to what it was. The wiggle's amplitude is what buys the length and the number of wiggles is free, so all three curves have exactly the target length while the third stays 16 times closer to the shortened segment than the first. That is the mechanism: the family converges to a map that is not isometric, while every member of it is.

Impossible in two derivatives, possible in one

The impossibility this whole collection rests on computes a second derivative, so it is a statement about maps that have two. Take one away and it is false: a corrugation restores an exact length while converging to the map that does not, and iterating it gives a flattening whose derivative converges and whose curvature runs to half a million.

impossibility · Curvature The icosahedron's faces, drawn on the sphere. The edges of the icosahedron projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 25.8° from its centre to its own boundary, and the gnomonic map onto it reaches 6.6° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion.

The globe on a solid

Cylinder, cone and plane are not the only surfaces a sphere can be laid on. Project it onto a polyhedron and the curvature goes entirely to the corners — π at each of the tetrahedron's four, π/5 at each of the dodecahedron's twenty, and always 4π in total, which is exactly the curvature of the sphere it replaced.

families · Polyhedral One construction, one number. A ray from a point on the equatorial plane, through the sphere, onto a tangent cylinder. Where the centre sits is the whole of the family: on the axis it gives tan φ, at the far side of the sphere 2 tan(φ/2), and at infinity sin φ. The same expression with angular distance in place of latitude, and a tangent plane in place of a cylinder, gives the gnomonic, the stereographic and the orthographic. The surface decides whether the answer is read as a height or as a radius and nothing else.

The developable surface was never necessary

Eight essays on this ladder start from cylinders, cones and planes, and none of them has asked which named projection is actually a projection onto one. Fitting a one-parameter perspective construction to eleven of them: six come back exact, and the three landmarks are the gnomonic at h = 0, the stereographic at h = 1 and the orthographic at h = ∞.

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

Named alongside it

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

ConeGaussian curvatureAzimuthalCone constantCylinderCylindricalIntrinsic geometryIsometryTaxonomyTheorema EgregiumConicAspect

All concepts