Zoom is a ladder

The scale of a screen map is not one number shows what latitude does to a printed scale: a zoom level advertises one denominator for the whole world and the map is at that denominator along exactly one line.

There is a second quantisation and it arrives before the map is drawn. A tiling scheme exists only at integer zoom levels, and those are a factor of two apart in linear resolution, so a request for any scale between two rungs is answered with whichever rung is nearer.

The rungs, and where they came from #

The pyramid’s zero level is one tile of 256 pixels covering the whole projected world, so its resolution is $2\pi R/256 = 156{,}543$ metres per pixel at the equator. Every level halves it.

The denominators are what they are because 256 and 2 and the Earth’s radius are what they are. Not one of them is a round number, and not one of them is a scale any cartographic tradition ever used.

The mismatch is √2 and cannot be otherwise #

A ladder whose rungs are a factor of two apart, with a nearest-rung rule, puts every request within a factor of $\sqrt{2}$ of a rung. That is arithmetic rather than measurement, and measuring it is a check on the rule rather than on the numbers: the sweep above reports 1.4125 and 0.7075 across an octave, against $\sqrt2 = 1.41421$ and its reciprocal, and the small shortfall is the sweep’s own sampling.

In area the same mismatch is a factor of two, because a linear ratio squares. A reader who asked for a map at 1:25,000 and got 1:34,124 is looking at a picture with 1.86 times as much ground in each pixel as requested.

The worst case is the interesting one and the typical case is not much better. Averaged over an octave, the ratio’s mean absolute departure from one is 17.5 per cent linear and 36.1 per cent in area, so a randomly chosen requested scale is typically a third wrong in area.

The traditional scales all land badly #

Every cartographic tradition uses round denominators — 1:1,000, 1:2,500, 1:10,000, 1:25,000, 1:50,000, 1:250,000, 1:1,000,000 — and none of them is a power of two times anything.

The two worst are 1:25,000 and 1:50,000, both 36 per cent coarser than asked, and both are the workhorse scales of national topographic mapping. The best is 1:1,000, at 6.6 per cent finer.

There is a pattern in the ratios that is worth reading: 1:25,000 and 1:50,000 have the same ratio, as do 1:10,000 and 1:2,500, and 1:250,000 and 1:1,000,000. Any two requested scales in a ratio of a power of two land at the same place on the sawtooth, which is the periodicity restated — and it means a national series’ whole ladder of scales inherits at most a handful of distinct mismatches.

Latitude moves it as well #

The rung a request lands on depends on the latitude, because the ground resolution of a tile does.

At the equator a request for 1:25,000 lands on zoom 14 and is served at 1:34,124. At 55° north the same request lands on zoom 14 as well but that level is worth 1:19,573 there, so the ratio is 0.783 rather than 1.365 — the map is now 22 per cent finer than asked instead of 36 per cent coarser.

So the two quantisations compound. The published denominator is wrong by $\sec\varphi$ and the rung is wrong by up to $\sqrt2$, and a reader who wants to know what scale is on the screen has to correct for both.

Over-zooming, which is the same ladder read downwards #

A renderer with no tiles at the level it wants serves a lower one scaled up. The picture arrives at the right size and its pixels are worth more ground than they claim.

That is a different failure from the ladder’s own quantisation and it is worth separating. The ladder’s mismatch is a choice of rung and delivers a real map at a real resolution; over-zooming delivers a map at a resolution the interface denies. A reader can compensate for the first by knowing the rule and cannot compensate for the second at all, because the missing detail was never in the tile.

The rung a print job lands on #

A screen map printed at a stated size inherits everything above and adds one more quantisation, because a printer’s resolution is a third ladder with rungs of its own.

The chain is: the reader asks for a scale; the pyramid supplies the nearest rung, up to 41 per cent away; the latitude moves the rung’s true scale by $\sec\varphi$; and the print device resamples the pixels to its own dots. Three quantisations, and only the last is ever stated on the output.

That is why the scale bar on a screen map is the honest object and the printed ratio is not. A scale bar is right in one place measures the bar’s own failure — it is a picture of a distance and is true along one line — but the bar at least scales with the picture through every one of the three steps above, which a printed denominator does not.

What a specification should ask for #

The consequence for anybody writing a requirement is that a scale is the wrong unit to write it in.

A requirement of “1:25,000 or better” is satisfied by zoom 15 and violated by zoom 14, and the gap between them is a factor of two, so the requirement’s true meaning is “zoom 15”, with a 46 per cent margin nobody intended. A requirement stated in ground resolution — “0.5 metres per pixel or better at 52° north” — names a rung directly and inherits none of the ladder.

That is the same move the pixel is a place with a size makes for position: state the requirement in the units the scheme quantises in, and the quantisation becomes a known step rather than a hidden one.

What the ladder buys, which is why it is there #

A ladder of powers of two is not an accident of implementation. It is what makes the pyramid a quadtree: each tile splits into exactly four at the next level, every tile’s ancestors are computable by integer arithmetic, and a request for any area at any level touches a bounded number of tiles.

A screen map is a pyramid of tiles works through what that structure forces, and the scale quantisation is one of the things it forces. A scheme with continuous scale would need a tile for every scale, which is not a scheme.

So the trade is: continuous scale, or a cacheable hierarchy with a bounded number of tiles per view. Every slippy map in the world took the second, and the sawtooth in this essay’s first figure is the price, printed nowhere.

Two of the scheme’s other constants come out of the same decision. The world has to be square for the quadtree to work at all, which forces the cut at 85.0511° and drops 1.9 million square kilometres — the square costs the poles measures that, and finds the usual defence overstates the requirement by a factor of 534. And the tiles have to be independent for the caching to work, which produces the clipping this site measures in a tile is drawn without its neighbours. Three consequences, one design, and the scale quantisation is the mildest of them.

The generalisation ladder is the real one #

The scale mismatch matters less than it sounds, and saying why is the honest way to size it.

A map is not only a scale. It is a selection: which features appear, which are dropped, how lines are weighted, where labels go. That selection is made per zoom level, so a tile pyramid has a discrete ladder of generalisations as well as of resolutions, and the two are locked together.

The consequence is that asking for 1:25,000 and being given zoom 14 does not deliver a 1:34,124 map. It delivers the cartography somebody drew for zoom 14, at whatever size the screen is. A 36 per cent scale mismatch is a rounding beside a whole level’s difference in what is drawn.

That is a genuinely different structure from a paper series, where each sheet’s generalisation was drawn for its own scale and the reader could not change it. A screen map’s reader changes scale continuously and the generalisation changes in steps, so the mismatch a reader actually sees is a picture drawn for one scale and displayed at another — which is the same complaint, one level up.

The intermediate scales that do exist #

Modern renderers do draw at fractional zoom, by scaling the tiles of the nearest integer level — which is over-zooming by a fraction of a level rather than by a whole one, and produces a picture at the requested scale with the detail of the nearest rung.

That does not remove the quantisation; it moves it from the scale to the detail. The map at 1:25,000 drawn from zoom 14 tiles has the right size and the generalisation of 1:34,124, so its labels, its line weights and its selection of features are all those of a smaller-scale map, blown up.

Which of those two failures a reader prefers is a cartographic question rather than a geometric one, and the site does not take a view. What it can say is that both are consequences of the same ladder, and that a renderer offering fractional zoom has not escaped it — it has chosen which half of the mismatch to absorb.

Vector tiles change the arithmetic without changing the structure. A vector tile carries geometry rather than pixels, so it can be drawn at any size without resampling, and the resolution ladder stops applying to the rendering. What does not stop applying is the ladder of what each tile contains: a vector tile is still built per level, still drops features per level, and still simplifies its geometry to a per-level tolerance. The pyramid’s rungs move from the pixels to the content and the count of them is unchanged.

What a reader can do with the number #

The ladder is knowable, which makes it a different kind of defect from most of the ones this site measures.

A reader who wants to know the true scale of what is on the screen needs three things: the zoom level, the latitude, and the pixel size the scheme assumes. All three are available — the level from the interface, the latitude from the map itself, and the pixel size from the standard — and the arithmetic is

$$\text{denominator} = \frac{2\pi R\cos\varphi}{256 \cdot 2^{z} \cdot p}$$

with $p$ the assumed pixel, conventionally 0.28 millimetres. Everything in this essay is that expression evaluated at integer $z$.

The contrast with the site’s other measurements is worth drawing. The pixel is a place with a size measures a quantisation whose effect depends on where the tile grid happens to fall, so it is a rate rather than a value; this one is exactly determined by three known integers. A defect that can be computed is a defect that can be corrected, and the reason it is not corrected is that no interface offers the three numbers together.

What a different base would have cost #

The rungs are a factor of two apart because a tile splits into four. A scheme could have split into nine, giving rungs a factor of three apart, and the trade is worth stating because it says what the factor of two is buying.

With a base of three the worst mismatch would be $\sqrt3 = 1.732$ rather than 1.414, and the number of levels needed to span the same range of scales would fall from twenty to thirteen. Fewer levels means a shallower pyramid, fewer distinct cartographies to draw, and coarser control over what a reader sees.

With a base of $\sqrt2$ — every other level a half-step — the worst mismatch would be 1.19 and the level count would double. That is the scheme somebody would design if the mismatch were the thing being minimised, and nobody has, because the level count is what costs money: every level is a full rendering of the world.

So the factor of two is a choice about production rather than about geometry, and the 41 per cent is what the choice costs the reader. The pyramid did not have to be Mercator makes the same kind of argument about the projection — the scheme’s constants are trades rather than necessities, and each one is a decision somebody made once.

A constant chosen for production, paid for by the reader #

The factor of two is the last of the pyramid’s constants to be priced, and the three together make a pattern worth naming.

Each constant was chosen for a production reason. The tile size for bandwidth, the projection for a square that quarters cleanly, the zoom base for a quadtree that halves. Every one of those is a sound engineering decision about how images are made, stored and delivered.

And each is paid for by the reader in a cartographic currency. The tile size decides what a drawn mark means on the ground; the projection decides that the poles are absent and that scale varies by a factor of eleven; the zoom base decides that a reader asking for a scale gets one up to 41 per cent away from it.

None of the three is visible as a choice from the reader’s side. The interface offers zoom levels, not scales; a sheet, not a projection; a picture, not a stroke width in metres. So a reader has no way to know that any of the three was decided, let alone that it was decided against a criterion that had nothing to do with them.

And the pattern says where to look in the next such system. Not at the parts anybody designed as cartography, which get argued about, but at the constants adopted for delivery — the chunk size, the addressing scheme, the refinement ratio, the cache key. Each of those is chosen early, by people solving a different problem, and each ends up deciding something a reader experiences as a property of the map.

Which is the whole argument of this ladder in one sentence. A delivery system’s constants become a map’s properties, the two kinds of reason are never weighed against each other because they are held by different people, and the only way to see the trade at all is to compute what the reader’s side of it costs.

The reader never sees any of them being chosen, which is why the pricing has to be done from outside.

Nothing in the interface has a place to record that such a trade was made,Nothing in the interface has a place to record that a trade was made, so the reader’s side of it is invisible to everybody including the people who made the choice.

Where this ladder goes #

The screen field has established the pyramid as a coordinate system, the scale that is not one number, the bar that is right in one place, the pixel with a size, the cut that squares the world, the tile drawn without its neighbours, and the pyramid that did not have to be Mercator.

This adds the quantisation of scale itself, which sits before all of those: the reader does not choose a scale, they choose a rung, and the two differ by up to 41 per cent before latitude is considered.

What is left in this field is the other axis of the pyramid — time — where the same quantisation appears with a different constant, and where a tile’s age rather than its resolution is the thing an interface does not report.