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

Cone — where it appears

One of the three developable surfaces the classical account of projections is built on, touching or cutting the sphere along one or two parallels. Its apex angle is the cone constant, and letting that constant run from one to zero turns the azimuthal family continuously into the cylindrical one.

Named by 7 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 Six surfaces and their Gaussian curvature. Curvature computed at the centre of each surface, by both available routes — from the way the surface sits in space, and from distances measured inside it alone. The two agree, which is Gauss's theorem. A surface with K = 0 can be unrolled flat without stretching: here the plane, the cylinder and the cone do, and the sphere, the torus and the pseudosphere do not.

No map is faithful

Not "no map yet", and not "no map at page size". Gaussian curvature can be computed from inside a surface, a sphere has some and a plane has none, and that closes the question permanently.

impossibility · Curvature 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 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.

Developable surfaceCylinderTaxonomyAzimuthalCone constantConicCylindricalGaussian curvatureAspectIntrinsic geometryIsometryStandard parallel

All concepts