Giving up continuity

Every projection elsewhere on this site is a continuous map of the sphere onto the plane, and every one of them is forced to distort because a continuous map has to absorb the whole region’s curvature.

There is a way out, and it is the only one that genuinely works: stop requiring the map to be continuous.

Why it works #

The mechanism is the scale rule, applied per lobe.

A pseudocylindrical projection’s shape distortion is largely a function of distance from its central meridian: on a sinusoidal projection the meridian scale is $\sqrt{1 + \lambda^2\sin^2\varphi}$, which is one on the central meridian and grows quadratically away from it.

Cut the world into $n$ lobes and give each its own central meridian, and no point is ever more than $180°/n$ from one. The distortion each lobe has to carry is the distortion of a strip rather than of the whole sphere, and a strip’s total curvature is a fraction of a cap’s.

Each lobe is the same projection, rotated in longitude and translated in the plane. Both are isometries of their respective spaces, so every distortion measure at a point is exactly what the uninterrupted projection has at the corresponding point — the site asserts that, and it holds to 2×10⁻⁸.

Which is the important structural fact: interruption does not improve the projection. It changes which part of the projection each place gets.

The benefit, measured #

The equal-lobe family behaves exactly as the mechanism predicts: each doubling of the lobe count roughly halves the mean angular deformation, because it halves the maximum distance from a central meridian and the distortion is quadratic in that distance.

The site asserts that monotonicity — more equal lobes must mean less distortion — and the assertion is worth having because the benefit is the half of the trade that gets drawn.

The lobe count is not the variable #

The Goode-style row in that table is out of order, and the reason is the most useful thing in it.

Six lobes, and it distorts more than four. Not because six is worse than four but because those six are not equal: the scheme keeps landmasses whole, so one of its northern lobes spans 220° of longitude and the four-lobe scheme’s widest is 90°.

The quantity that predicts the distortion is the width of the widest lobe, not the number of lobes. A scheme with twenty lobes, nineteen of them narrow and one covering half the world, is barely better than an uninterrupted map.

That is worth knowing because lobe count is what gets quoted. Goode’s homolosine is described as six-lobed, and the description implies a benefit the arrangement does not deliver — the arrangement was chosen to keep continents whole, which is a different objective and a defensible one, and it costs shape fidelity relative to an even cut.

The cost, measured #

Interruption’s cost is usually described and not quantified, and the quantification is the point of this essay.

The site measures the tear at each cut: take two points a tenth of a degree apart across a cut — about five kilometres on the ground at 60° — and measure how far apart their images land, as a fraction of the map’s width.

On the Goode-style scheme, sixteen of the eighteen cuts sampled separate neighbouring points by more than two per cent of the map’s width, and the worst separates them by eighteen per cent. Two places five kilometres apart, drawn a fifth of a map apart.

That is asserted rather than reported. If no cut ever separated neighbouring points measurably, the map would be continuous, which would mean the lobes had been placed wrongly and the improvement was illusory. It is the rare case of an assertion that requires a failure of the thing being drawn.

What the tear costs in use #

Three specific things stop working, and they are not the same three for every map.

Distance across a cut has no meaning. Not “is inaccurate” — has no meaning. The two points are drawn in different pieces and the distance between the drawings is a fact about the layout.

Direction across a cut has no meaning, for the same reason, and neither does area of any region a cut passes through.

Contiguity is broken. A region spanning a cut appears as two regions. On a land-centred interruption that is fine for continents and fatal for oceans: the Atlantic and Pacific are each cut into pieces, and any question about ocean circulation, shipping or sea ice is unanswerable on the map.

Which is why the interruption scheme is a statement about what the map is for. A land-centred cut ruins the oceans; an ocean-centred cut ruins the continents; and there is no arrangement that keeps both, because the two are complementary regions of a sphere and cutting one is not cutting the other.

That makes the interruption pattern the clearest case in the subject of a projection parameter that encodes a purpose. Every projection minimises something, and here the objective is written into the map’s outline where a reader can see it.

The equator is a cut too #

A detail that the essays which draw these maps do not mention, and the site’s own machinery found.

Most interruption schemes give the northern and southern lobes different central meridians, because land is distributed differently in the two hemispheres. Where they meet, at the equator, the two lobes are drawn about different centres — so the equator is itself a cut, with a visible kink in every meridian crossing it.

The site’s assertion that each lobe carries the uninterrupted projection’s distortion exactly had to be told to skip the equator for schemes of that kind, because a central difference taken across it straddles two lobes and measures the seam. That is the correct behaviour and it made the seam explicit.

An interrupted world map therefore usually has one more discontinuity than its lobe count suggests, running along the line most readers would assume is continuous. It is visible in any Goode homolosine once the kink is looked for, and it is never in the caption.

The gores that came first #

Interruption is older than flat maps that use it, and the older application makes the geometry obvious.

A globe is covered by pasting printed gores onto a sphere: long tapering strips, typically twelve or eighteen of them, each spanning 30° or 20° of longitude and running pole to pole. Each gore is printed flat and then curved onto the ball.

That is the same construction with the arrow reversed. A gore is a narrow lobe, printed in something close to a sinusoidal projection about its own central meridian, and it is narrow for exactly the reason an interrupted map’s lobes are narrow: a narrow strip can be flattened with very little distortion, so a gore can be printed flat and stretched onto a sphere without visible error.

Twelve gores of 30° each have a maximum distance from a central meridian of 15°, which by the numbers in this essay corresponds to a mean angular deformation of well under three degrees. That is why globe-making works at all, and why it needs so many strips.

The globe-maker’s problem and the interrupted-map maker’s problem are the same problem with the sign of the distortion reversed, and both are solved by the same observation: distortion scales with the area of the piece, so cut the piece down.

Cutting in three dimensions #

The idea has a large modern application that has nothing to do with maps, and it is worth naming because the trade is identical.

A three-dimensional model in computer graphics carries a texture, and applying a texture means finding a map from the model’s surface to a flat image. That is a projection problem with the same obstruction: a curved surface cannot be flattened without distortion, so the surface is cut into charts and each is flattened separately.

The design decisions are the ones in this essay. More charts means less distortion per chart and more seams. Seams are visible as discontinuities in the texture, so they are placed where nobody looks — under an arm, along a hairline — exactly as Goode’s cuts are placed in the ocean.

The vocabulary differs and the arithmetic does not. A UV unwrap is an interrupted projection whose interruption scheme encodes what the model is for, and the person doing it is making the trade this essay measures, usually without knowing that a cartographer made it first.

Goode’s homolosine, specifically #

The best-known interrupted projection is worth describing precisely because it combines two ideas.

Goode’s homolosine, from 1923, is equal-area throughout and is assembled from two different projections: sinusoidal between about 40°44′ north and south, and Mollweide poleward of that, joined at the latitude where their parallel spacings match. Both are equal-area, so the composite is too — exactly, everywhere, because equal-areaness is a pointwise property and each piece has it.

That is a second kind of discontinuity: not a cut in the map, but a change of formula at a latitude. The derivative is discontinuous across the join even though the map is not, so the graticule has a slight kink there as well.

Then the lobes: interrupted through the oceans, arranged so each continent stays whole. The result preserves area exactly, keeps shape distortion much lower than any uninterrupted equal-area projection, and is torn in six places plus the equator plus two latitude circles where the formula changes.

It is still the standard choice for a global thematic map of land, and the reason is that all of those defects are in the ocean and all of the accuracy is on the land, which is exactly what the map is for.

Why the equal-area property survives exactly #

A structural point that is easy to state and easy to get wrong in the other direction.

An interrupted equal-area projection is equal-area exactly, everywhere it is defined. Not approximately, and not on average. The reason is that equal-areaness is a pointwise property: it says the areal factor is one at every point, and each lobe is the base projection composed with a rotation and a translation, both of which have areal factor one.

Compare that with a property defined globally. If the projection had to preserve, say, the total area of the mapped region as a single number, cutting the region would raise a real question about what happened to the material at the cuts. It does not, because the property is local and every point is in exactly one lobe.

The cuts themselves are measure zero, so they contribute nothing to any area. The map is undefined on a set of zero area and exact everywhere else, which is a perfectly respectable state for a measure-preserving map to be in.

That is why interruption is available for equal-area projections at essentially no cost to the property, and it is why Goode’s homolosine can combine two different projections and stay exactly equal-area: both pieces have the property pointwise, so the composite does too.

When to cut #

The rule that follows, and it is narrower than the numbers suggest.

Cut when the map is about the pieces rather than about the whole. A thematic map of land, of biomes, of population — the questions are within continents and the cuts are in water nobody is asking about.

Do not cut when any question crosses the cut. Oceans, circulation, trade routes, anything global.

Do not cut a map that will be read as a picture of the world. The tears are severe and a reader who has not been told they are deliberate will read them as an error, which is a communication failure rather than a geometric one.

And the number of lobes should be as few as the accuracy requires, because each one costs a discontinuity and the benefit is in the widest lobe rather than in the count. Four is usually enough; eight is almost always too many.

An interruption is one way out of the family’s central trade. The other is the pole line, and it is worth seeing what the interruption is an alternative to.

What was computed here #

Every distortion figure is measured from the base projection’s own derivatives at the rotated position, and each lobe’s equivalence to the uninterrupted projection is asserted rather than assumed — a lobe is the same map rotated and translated, and it must measure identically, which it does to 2×10⁻⁸.

That tolerance is not zero and the reason is the same near-conformal cancellation that sets the whole site’s noise floor: on a central meridian a sinusoidal projection has $h = k$ exactly, so $\omega$ is the difference of two nearly equal square roots and a perturbation of $10^{-16}$ in the inputs surfaces as $10^{-8}$ in the output. Away from the central meridians the two routes agree to $10^{-13}$.

Three assertions carry the essay. More equal lobes must reduce the mean angular deformation, and each doubling must gain at least a quarter. The uninterrupted map must have no tear and the interrupted ones must have one. And the tears must be large — at least two per cent of the map’s width at some cut — because an interrupted projection that turned out to be continuous would mean the lobes had been placed wrongly.

What the pictures cannot show #

Both halves at once. The map figures show the benefit — a graticule that stays close to a central meridian everywhere — and the cost appears only as gaps at the edges of lobes, which read as the map’s outline rather than as tears. The trade-off figure shows both as curves and shows neither as a map.

The figures also cannot show what is on the other side of a cut, because that is the definition of a cut. Every question a reader might have about the relationship between two lobes is one the picture is unable to address, and that is the property rather than a limitation of the drawing.

Two seams nobody chose #

An interruption is a tear placed deliberately, where the ocean can absorb it, with its size measured. The applied field meets two seams of a different kind: one forced by arithmetic and one chosen for delivery.

Longitude is a coordinate on a circle and the numbers are a coordinate on an interval, so the numbering has a cut somewhere and every choice of cut leaves some ground crossing it. And a tile scheme’s boundaries are seams too — a feature spanning nine tiles is nine features to a renderer, labelled at nine centroids up to 1,863 kilometres from the right one.

The first cannot be moved away, only moved. A shape crossing the antimeridian has a bounding box eighteen times too wide, a planar area seventeen times too large, and a midpoint at the antipode of where it belongs — none of which is a matter of degree, and all of which follow from a tear that no cartographer chose.

The second is chosen, and its pieces are exact: their areas sum to the whole to within 2 × 10⁻¹⁶. What is not exact is anything computed per piece — the centroids of the parts are not the centroid of the whole, and a renderer that labels each part has produced a label at a place nothing is — a tile is drawn without its neighbours.

Who found it, and when #

Interrupted projections are older than they look. Ptolemy’s second projection is not interrupted, but sixteenth-century globe gores — the printed strips that were pasted onto a sphere — are interruption in the other direction, and the geometry is the same.

The systematic use of interruption for flat maps is late nineteenth and early twentieth century. Goode published the homolosine in 1923 at the University of Chicago, explicitly as an alternative to Mercator for classroom use, and it has been the standard equal-area world map for thematic land mapping since.

The idea has had a quiet revival in computer graphics, where a sphere is routinely unwrapped into charts for texture mapping and the choice of where to cut is the same problem with the same trade — small distortion per chart against the number of seams, decided by what the seams will run through.

Where this goes next #

The projections that give up exactness instead of continuity are compromise projections. The bound that explains why smaller pieces distort less is total curvature and the scale rule. And the projection whose discontinuity was not declared in advance is the projections that gave up being one thing.