The pyramid did not have to be Mercator

Ask why every screen map is on Mercator and the answer comes back in one of two forms. The historical one — it was there, it was familiar, the first schemes used it and everything after copied them — is true and is not a reason. The technical one is that a quadtree of square tiles needs a square world and Mercator gives one.

The technical one is false as stated, and the projection that refutes it takes one line to derive.

The projection that also gives a square #

A cylindrical equal-area projection with standard parallels at $\pm\varphi_0$ is

$$x = R\lambda\cos\varphi_0, \qquad y = \frac{R\sin\varphi}{\cos\varphi_0}$$

which maps the sphere onto a rectangle $2\pi R\cos\varphi_0$ wide and $2R/\cos\varphi_0$ tall. Those are equal exactly when

$$\pi\cos^2\varphi_0 = 1 \qquad\Longrightarrow\qquad \varphi_0 = \arccos\left(\pi^{-1/2}\right) = 55.6540°$$

One number, forced by the same requirement that forces Mercator’s cut, and arrived at by the same kind of arithmetic. The site’s gate asserts the resulting aspect ratio is one to twelve decimal places, because a standard parallel solved loosely would still give a nearly square world and would make everything downstream unearned.

That projection is not exotic. It is a member of the one-parameter family what a standard parallel buys works through — the family that contains Lambert’s at 0°, Behrmann’s at 30° and Gall–Peters at 45° — and this member is the one nobody named because nobody was looking for a square.

And it needs no cut #

The equal-area world’s northing is $\sin\varphi/\cos\varphi_0$, which is bounded. At the pole it is $1/\cos\varphi_0 = 1.7725$, and the pole is on the map: not as a point but as the top edge, a line as long as the equator.

So the 1,901,487 square kilometres that the square costs the poles computes as the Mercator scheme’s discard is not discarded here. The equal-area pyramid carries the whole sphere, exactly, with no truncation latitude to explain and no polar data centre needing a separate scheme.

What it costs, measured #

Every gain on this site is stated with its price attached, and the price here is angular deformation — the invariant $\omega$, computed from the projection’s own derivatives rather than described.

Two columns are exactly zero and two are exactly one, which is the two-sided assertion this site has used since foundation: each projection is required to be perfect in its own property and to fail in the other, so a version of either that had quietly become the identity would be caught.

The number in the middle of the table is worth pausing on. At 55.654°, where the equal-area scheme is undistorted, Mercator’s areal factor is exactly π. That is not a coincidence and not a numerical accident: the square-world condition is $\cos^2\varphi_0 = 1/\pi$ and Mercator’s areal factor is $\sec^2\varphi$, so the two statements are the same equation. The machinery found it before anybody looked for it, and it survives to 3.1416 in the measurement.

What conformality actually buys #

If the square is not the reason, the reason must be conformality, and it is worth being precise about what that is worth in an interactive map rather than in a textbook.

A rotated view does not shear. Screen maps rotate now — with a heading, with a gesture, in a vehicle. On a conformal map the picture under rotation is the same picture turned; on an equal-area map at 60° north, with $\omega = 13.8°$, a north-up view and a north-east-up view of the same street are visibly different shapes.

Imagery overlays. Aerial and satellite imagery is orthorectified onto the same projection as the vector data. On a conformal map a building is a rectangle at any latitude; on the equal-area one it is a parallelogram whose shear angle is a function of where it is, so a single style, a single icon set and a single set of sprite images could not be correct everywhere.

The pixel is isotropic. This is the quiet one and it may be the decisive one. On a conformal map $h = k$, so one pixel covers the same ground north–south as east–west and metres per pixel is a single number. On the equal-area scheme a pixel’s ground footprint is a rectangle whose east–west to north–south ratio is $h/k = (\cos\varphi/\cos\varphi_0)^2$: π:1 on the equator, 1:1 at the standard parallel, and 1:4.8 at 75°. Every resolution figure, every tolerance and every symbol size would carry a direction as well as a latitude.

That last point converts an aesthetic preference into an engineering one, and it is the honest answer to why Mercator: not the square, and not the tradition, but that a conformal projection is the only kind on which a pixel is a square of ground and a circle drawn on the screen is a circle on the Earth.

The resolution table it would have published #

The isotropy argument has a table behind it, and the table is the most concrete thing in this essay.

The equal-area world is $2R\sqrt\pi$ across, so at level 0 its pixel is 88,320 projected metres against Mercator’s 156,543. On the ground that pixel covers

$$\text{east–west } \frac{P}{k} = P\,\frac{\cos\varphi}{\cos\varphi_0}, \qquad \text{north–south } \frac{P}{h} = P\,\frac{\cos\varphi_0}{\cos\varphi}$$

and since $hk = 1$, the two move in opposite directions. At the standard parallel both are 88,320 metres and the pixel is a square of ground. At the equator they are 156,543 and 49,829 — a footprint $\pi$ times wider than it is tall, which is the square-world condition showing up for the third time in this essay. At 75° they are 40,514 and 192,540, a footprint 4.75 times taller than wide.

So the scheme’s published table would need two columns of ground resolution rather than one, both functions of latitude, and every specification written against it — imagery resolution, simplification tolerance, symbol size, label spacing — would need a direction attached. The Mercator table needs one column and a cosine, which is the version this field spends two essays complaining about, and it is the simpler of the two by a wide margin.

Why nobody will switch, which is not a geometric reason #

The arithmetic above says the equal-area pyramid is a coherent design. Nothing in it says the choice could be revisited, and the obstacles are worth naming so that they are not mistaken for arguments.

Every cached tile becomes wrong. The world’s imagery has been rendered, cached and distributed against one projection; changing it invalidates all of it at every level.

Every scale rule becomes wrong. A style that shows a layer between two zoom levels has been tuned against Mercator’s ground resolution at the latitudes its authors work in. On the new scheme those thresholds move by up to a factor of two, in a direction that depends on latitude and on whether the rule was about width or height.

Every overlay misaligns. Anything drawn by a second system in Web Mercator coordinates — an annotation, a heat map, a third-party layer — is in the wrong place until it is reprojected, and the second half of this field measures what reprojection costs an image.

None of that is geometry. It is the same lock-in that keeps a national grid on a datum fitted in the 1930s, described in a datum is fitted to a region, and it deserves the same treatment: state it as an inheritance rather than as a justification, so the next scheme is chosen on its merits.

The comparison the field is for #

Two schemes, both square, both quadtrees, and a clean statement of what each is good for. That is purpose before property in its strongest form: neither is better, and the question which projection is best has no answer until the operation is named, which is the whole argument of which projection is best.

The equal-area pyramid is better for: any thematic map whose shading is a density; any comparison of extents across latitudes; showing the poles; and any measurement of area taken directly off the picture, which computing an area needs a surface shows is otherwise wrong by a factor of three over a mid-latitude cell.

The Mercator pyramid is better for: rotation, imagery, isotropic pixels, symbol geometry, navigation — a rhumb line is straight, which is why Mercator exists — and any local view, where its area error is negligible and its shape fidelity is exact.

The second list is what a general-purpose interactive map does all day and the first is what a thematic map does once. The choice made was the right one. It was not made for the reason usually given, and the reason usually given is checkable in ten lines of arithmetic and wrong.

The projection nobody would choose, which is also square #

There is a third candidate worth naming because it makes the constraint concrete: the plate carrée, whose world is $2{:}1$ and which therefore needs a two-tile root rather than a square. It is neither conformal nor equal-area — its areal error reaches a factor of two at 60° and its angular deformation reaches 38.9° — so it fails both columns of the table above.

It is nonetheless the second most widely tiled projection there is, because a coordinate on it is the coordinate in the file: $x = \lambda$, $y = \varphi$, with no transformation at all. That is a real advantage of a real kind — no reprojection means no resampling, which the second half of this field measures — and it is bought by being mediocre at everything a map is looked at for. The plate carrée is the essay on the projection nobody chooses and everybody uses, and its appearance as a tiling scheme is the same story one layer down.

What would have to be true for the choice to change #

The trade is not eternal, and the conditions under which it flips are worth stating because they are all technological rather than mathematical.

If rotated views stopped mattering, the shear cost would fall. If imagery were reprojected per-view on the client — which is now cheap, and which the second half of this field measures the cost of — the orthorectification argument would weaken. If symbol geometry were computed per-latitude rather than drawn from a sprite, the isotropy argument would weaken too.

None of those is fanciful. What would not change is the arithmetic: the square-world equal-area projection would still be at $\pm55.654°$, its deformation would still be 13.8° at 60°, and Mercator’s areal factor would still be four there. The measurements outlive the trade, which is the reason for making them numbers rather than adjectives.

The third option, which changes what the question is #

There is a way out of the trade, and it is the direction screen mapping has actually taken: stop baking the projection into the pictures.

A scheme that ships coordinates rather than images asks the reader’s own machine to project them at the moment of drawing. The tiling is then only a partition — a way of deciding which features travel together — and the projection it partitions by need not be the projection anything is drawn in. A client can draw the same tiles on Mercator, on an equal-area cylindrical, on a globe, or on a tilted perspective, and the pyramid is indifferent.

That does not abolish the trade; it moves it. The partition still wants a square world and a cheap forward map, and Mercator is as good a choice as any for that because nobody looks at it. The display projection is now a per-view decision, so it can be conformal while the reader pans and equal-area when the reader asks for a density map, which is the correct answer to a question this site keeps giving the same answer to: name the operation, then choose.

What it costs is that every vertex is transformed on arrival, at every frame, and that the client must carry an inverse for the partition and a forward for the display. Both are arithmetic on coordinates that were exact when they left, which is precisely the subject of the second half of this field: a straight segment is a claim about a plane, and a segment that was straight in the partition’s plane is not straight in the display’s.

When an interface cannot change, change the layer above it #

The reason nobody will switch is not geometric, and it is worth setting out properly, because the shape of it is what makes the third option the interesting one.

A tiling scheme became an interface, and an interface’s cost of change scales with how many independent parties implement it. Changing the pyramid’s projection is not a change to a projection. It is a change to the meaning of every tile URL in existence, every cached tile on every device, every zoom-to-scale table printed in every specification, every client library’s assumption that zoom 12 means a particular ground resolution, every third-party overlay aligned to the existing grid, and every offline package already downloaded.

None of those parties can move first and none can move alone. A provider switching projections serves tiles that no existing client can align; a client switching expects tiles no provider serves. That is a coordination problem rather than an engineering one, and coordination problems of this size are not solved by the new option being better.

Which is why the measurement in this rung is worth making anyway. It establishes what the choice cost, so that the next system built on a pyramid — a planetary body, a private tiling, a domain-specific scheme — is made with the number in hand rather than by copying what exists. The cost of the incumbent is not a reason to replace it; it is the design input for the next one.

It is worth noticing how narrow the escape is, because it does not generalise to every interface complaint. It works here only because the pyramid is doing two jobs that happen to be separable — deciding which data a tile holds, and deciding what the picture looks like — and because only the first of them is what anybody agreed to. An interface whose contract genuinely covers the thing to be changed offers no such split, and the coordination problem is then the whole problem.

And the third option escapes the coordination problem entirely, which is its real advantage. Separating the partition from the display changes nothing about the interface: the tiles keep their addresses, the zoom levels keep their meaning, and the client draws the same data in whatever projection it likes. No provider has to agree, no cache is invalidated, and a single client can adopt it unilaterally.

The split is also the reason this rung can be a measurement rather than a proposal: the number it produces is useful whether or not anybody ever acts on it.

That is a general pattern rather than a trick. An interface that cannot be renegotiated can often be left alone while the layer above it takes on the responsibility — the partition stays Mercator because addresses are Mercator’s, and the picture stops being Mercator because the picture is nobody’s contract. What it costs is the arithmetic on arrival and the loss of the coincidence that made the two the same thing.

Where this ladder ends #

Five rungs: the pyramid as a coordinate system, the scale it prints, the bar that shows the scale, the cut at the top of the world, the pixel, the tile’s ignorance of its neighbours, and finally the choice of projection that all of them inherit. Every one of them is a decision about delivering a picture that turns out to be a decision about geometry.

The other half of this field starts from the file rather than from the screen, and asks what happens when a program is asked a question about the coordinates rather than asked to draw them. The first thing it has to establish is that a coordinate on its own does not say what it refers to.