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

Continuity — where it appears

The property of a map that places close together on the ground stay close together on the page, with no jump anywhere. A projection written in a longitude jumps wherever the longitude wraps, so continuity as a function of the place is much rarer than it looks.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

How each projection escapes being one to one. Eighteen projections, and the theorem allows no fourth column. A map of the whole sphere either leaves ground undrawn, or draws one place as a curve, or is cut so that one ground curve appears twice — and 11 of the eighteen do more than one of those. The three columns are a solid angle, a count of places and a page length, which is why they are three columns rather than one score: there is no rate at which a hemisphere converts into a pole.

No map of the whole sphere is one to one

Eleven essays establish that no map preserves distance, and every step of that argument needs a distance. There is a second impossibility underneath it that needs nothing at all: a sphere is compact and has no boundary, so a continuous map of it into the page cannot also be one to one. Eighteen library projections, eighteen escapes, and not one of them free.

impossibility · Topology Ground north, on a map that has the pole on it. Every arrow points along its own meridian, towards the pole, and the pole is at the centre. Walking once anticlockwise round any loop enclosing it turns the arrow once anticlockwise as well: the index is 1, counted as a winding number with no distance anywhere in the calculation. A field like this cannot be combed flat. There is no way to choose a page direction for north at every point of the neighbourhood without the choice tearing somewhere, and the somewhere is the point in the middle.

North cannot be up everywhere

Ground north is a field of arrows on the sphere, and a field of arrows on a sphere must vanish somewhere. The failure is not measured, it is counted: the indices of the zeros sum to two, obtained here as a winding number in seven different charts with no distance anywhere in the calculation, and it is the same two that Gauss–Bonnet gets by integrating curvature.

impossibility · Topology The map that is continuous, and the pair it pays with. The orthographic is defined and continuous at every place on the Earth — it is written in the components of the place itself, with no longitude in it to jump. What it gives up is being one to one, and it gives it up almost everywhere: 47 per cent of the sphere shares its page point with the place directly behind it. Borsuk–Ulam guarantees at least one ANTIPODAL pair among those, and here it is exactly one — the centre and the place on the far side of the world, both at the middle of the picture, found to a residual of 1.5e-14.

Two opposite places on the same spot

A projection may be continuous everywhere or one to one everywhere, and the first two rungs price both. What neither says is that the choice is not symmetric: a map that keeps continuity does not lose injectivity somewhere arbitrary. It loses it, always and at minimum, on a pair of places directly opposite each other on the Earth.

impossibility · Topology The world after a map of degree 2. The graticule of the sphere, sent through the square of the stereographic coordinate and then drawn in Mollweide. Every parallel is still a closed curve and every meridian still runs pole to pole, and the whole world has been wrapped round the sphere 2 times: the 2 meridians that used to be 180° apart now lie on top of one another. Nothing has been torn. The degree measured by integrating the area the map sweeps is 2.0004.

How many times, not whether

Three rungs of this ladder answer yes or no and have no other kind of answer. The degree is the first quantity here that counts: a continuous map of the sphere to itself covers it a whole number of times, injectivity forces that number to ±1, and the number is recoverable three ways — by counting preimages, by integrating swept area, and from the rate at which the preimages coalesce.

impossibility · Topology What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole.

What a cut buys

Cuts are usually counted and never measured. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.

impossibility · Topology The corner an equal-area face map puts in a feature, and the one the gnomonic does not. A great circle crossing the seam between two faces, drawn on each face's own map and unfolded flat, with the angle between the incoming and outgoing tangents plotted against how obliquely it crosses. Under the equal-area face map every solid gives a corner: zero for a perpendicular crossing, where the two faces are symmetric about the edge, peaking near thirty degrees of obliquity and falling again as the crossing lies down along the edge. Under the gnomonic it is zero at every angle on every solid, which is the flat line on the axis.

The gnomonic crosses a seam without a corner

Choosing a polyhedral map means choosing a solid, a face map and where to cut. This one is about a seam the net does not cut, where two faces stay joined and carry two different maps — and the expectation was a corner in every feature crossing one. At the midpoint of the edge — which is where every crossing here is taken, and which turns out to be the one place on it a face's own mirror forces the corner to vanish — the gnomonic gives none at any obliquity on any Platonic solid, and the equal-area face map gives one of up to 0.39 degrees.

families · Polyhedral One map made of two, and the latitude where they meet. Goode's homolosine drawn without its interruptions: sinusoidal between 40°44′11.98″ north and south and Mollweide beyond, joined at the latitude where a parallel has the same length in both. Dashed: the join. Heavy: four meridians, drawn from 60° south to 60° north, which cross it. The central meridian crosses smoothly; the others turn a corner there, reaching 13.30° at 110° of longitude, marked. The map is equal-area everywhere, and every parallel keeps its length across the join.

Joined where the parallels agree, and the meridians turn a corner

Goode's homolosine is two projections joined at 40°44′11.98″, and three quantities match there: the length of the parallel, the areal factor, and the spacing of the parallels on the central meridian. The fourth does not. Every meridian but the central one turns a corner at the join, up to 13.30° at 110° of longitude, and the angular deformation drops 26° across a line of no width. Moving the join decides where that corner is paid, not whether.

families · Families The same composite with a strip spliced in, and the meridians no longer turn a corner. Goode's composite redrawn with an equal-area strip of 8° of latitude spliced between the sinusoidal and the Mollweide cap, shaded. The strip's parallel half-length is the unique cubic matching the sinusoidal's value and slope at 36.74° and the Mollweide's at 44.74°, so every meridian arrives and leaves along one direction and the 11.2° corner the plain join turns at 60° of longitude is gone. Nothing else about the map is changed: the cap is the unmodified Mollweide, and the composite is equal-area throughout.

An equal-area strip removes the corner and charges nothing for it

Goode's seam matches three quantities and breaks the fourth: every meridian but the central one turns a corner there. A third map spliced between the two removes it — four conditions on one function, satisfied uniquely by a cubic — and the expected price does not arrive. The spliced composite carries less angular deformation across the seam than the plain join, not more: 34.7° against 37.7° at sixty degrees of longitude. What it does leave is the same defect one derivative up, a jump in curvature falling as the reciprocal of the strip's width.

families · Families Goode's composite with a strip of a generalised Eckert II, whose meridians are straight. The sinusoidal to 32.28°, a strip of the generalised Eckert II projection with r = 0.8311 from there to 49.27°, and the unscaled Mollweide beyond, both hemispheres; the strip's edges are dashed. Inside the strip every meridian is a straight line, as Eckert II's are, and it leaves the sinusoidal's meridian and joins the Mollweide's along their own tangents, so the 11.9° corner Goode's join turns at 180° of longitude is gone. The composite is equal-area throughout, and its strip is 17.0° wide and centred at 40.78°, four hundredths of a degree from Goode's own join.

Only Eckert II's family can be the strip, and its meridians are straight

The cubic that smooths Goode's seam is four conditions on the length of a parallel, and equal area turns them into four conditions on how the parallels are spaced. The sinusoidal's spacing is flat, so a strip can leave it only where its own spacing is momentarily flat — and of eleven named equal-area maps, only Eckert II's spacing ever turns over. One member of its family meets all four conditions exactly, over seventeen degrees centred four hundredths of a degree from Goode's join, with every meridian straight. The cubic over the same span is the same map to four parts in a hundred thousand.

families · Families

Named alongside it

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

Equal-areaTopologyDiscontinuityGoode homolosinePseudocylindricalSeamSinusoidalVerificationAngular deformationInjectivityInterruptionInvariant

All concepts