The light comes from a page direction

The most-printed derived product in cartography is a shaded relief — a picture of a slope, and a slope is not a shape, and every implementation of it takes the same two numbers: an azimuth, conventionally 315°, and an altitude, conventionally 45°.

The azimuth is measured from the top of the sheet. It has to be: the shading is computed on the projected raster, after the terrain has been resampled into the map’s own coordinates, and the raster’s rows and columns are the page’s axes. The top of the sheet is grid north.

Grid north is not north. This collection established that four essays ago in another field, and never carried it into a derived product.

What the legend says and what the sheet does #

A shaded relief carries a note: illuminated from the north-west. A reader uses it to interpret the picture — a slope facing the light is bright, a slope facing away is dark, and the reader infers which way each hillside faces.

That inference is a compass inference and the light is not a compass direction. It is a page direction, and the two differ by the meridian convergence, which is zero on the central meridian and grows away from it.

The size, on an ordinary sheet #

On a Lambert conformal conic covering eighty degrees of longitude — a continental sheet, nothing exotic — the ground azimuth of a declared 315° light runs over 52.6 degrees.

Fifty-two degrees is a season’s worth of sun. It is the difference between a north-westerly light and a westerly one, and a reader identifying which slopes are lit is reading a different illumination at the two edges of the sheet.

Seventy-three per cent of the sheet is shaded measurably differently from what the legend describes, with a worst-case difference of 0.23 on a scale from zero to one — nearly a quarter of the full tonal range.

Which projections are immune, and why #

Exactly the ones with no meridian convergence: a cylindrical projection in its normal aspect, where every meridian is a vertical line and grid north is true north at every point.

That is a short list and it contains Mercator, the Lambert cylindrical, the plate carrée and Miller — none of which is used for a continental relief map, because their area distortion at the latitudes where mountains are is unacceptable.

So the projections that get the light right are the ones nobody shades, and the projections that get shaded — conics, for mid-latitude continents — are the ones that turn it most. That is not a coincidence: a conic’s cone constant is what makes it good for a mid-latitude band and is exactly what makes its meridians converge.

Where the top of the sheet comes from #

It is worth being precise about the chain, because each link is individually reasonable.

A terrain model is stored as a raster: a rectangular array of heights, with rows and columns. To draw it on a map it is resampled into the map’s projection, so that its rows run along the page’s horizontal and its columns along the page’s vertical. That resampling is the moment the array’s axes become the page’s axes.

The shading operator then reads the array. It computes a gradient from the differences between neighbouring cells, which gives a direction in array coordinates, and compares it against the declared azimuth, which is also in array coordinates because there is nowhere else for it to be.

Every step is correct and the composition is a page-frame calculation from beginning to end. The ground enters nowhere, and the legend’s claim about a compass bearing is added at the end by a person who has not been told.

It is not a small correction and it is not applied #

The correction is trivial. The meridian convergence is computable in closed form for every projection in the library; adding it to the declared azimuth at every raster cell would cost one extra term per pixel and would make the legend true.

No implementation does it. Every hillshade in general use — in desktop GIS, in raster processing libraries, in web rendering pipelines — takes one azimuth and applies it uniformly, because the algorithm operates on a raster and a raster has no idea what projection it is in.

That is the structural reason and it is worth naming. The shading operator is a neighbourhood operator on an array. It reads a three-by-three window and produces a number, and nothing in that description mentions a coordinate system. The projection is upstream, the legend is downstream, and the operator in the middle is geometry-blind.

The angle responsible is zero on the central meridian, grows away from it, and grows faster the higher the latitude — so a shaded sheet covering a wide band of longitude at high latitude is lit from the most different quarters at its two far corners, and the disagreement is worst exactly where the sheet is largest.

Two errors in the same product #

There is a second, better-known problem with hillshading a projected raster, and the two compound.

The slope computed from a projected raster is the slope on the page, so it is the ground slope divided by a scale factor that varies — the argument a slope is not a shape makes at the start of this ladder. That affects how steep everything looks.

This rung is about how it is lit, which is a direction rather than a magnitude, and it is the one that changes which slopes appear to face the viewer.

Together they mean that a shaded relief on a conic is wrong about the steepness by a factor that varies with latitude and wrong about the illumination by an angle that varies with longitude. Neither is corrected and neither is stated.

What a reader actually does with a hillshade #

The reason this matters more than a distortion figure suggests is what shading is for.

A contour map is read quantitatively, and the value on the contour survives the projection even when the reading does not: a reader takes a value off a line. A shaded relief is read qualitatively and almost pre-attentively — a viewer sees ridges and valleys without deciding to, and the seeing depends on the light being consistent.

An inconsistent light does not produce a visibly wrong picture. It produces a picture in which the same landform appears to face differently in different parts of the sheet, and the reader has no way to notice, because there is nothing on the page to compare against.

That is the same failure mode as the contour is right and the reading is wrong: the product is not broken, it is subtly and consistently misleading, and the misleading part is the part a reader trusts without checking.

Where the swing is largest #

The convergence on a conic is $n\,\Delta\lambda$, where $n$ is the cone constant and $\Delta\lambda$ is the longitude from the central meridian. So the swing is proportional to the sheet’s width in longitude and to the cone constant.

A cone constant near one — a conic fitted to high latitudes, approaching an azimuthal projection — swings the light degree for degree with longitude. A cone constant near zero — a conic approaching a cylinder — swings it not at all.

That gives a rule of thumb with a real derivation behind it. The light swings by about the sine of the central latitude, times the longitude range. At 45° that is 0.71 degrees per degree of longitude, and eighty degrees of longitude gives fifty-seven — close to the fifty-three measured, with the difference being the conic’s two standard parallels rather than one.

The aspect a reader recovers #

There is a way to see how large the effect is that does not involve tones at all, and it is the one a quantitative user runs into.

Many products derive an aspect raster alongside the hillshade — the compass bearing each cell faces — and it is used for slope-stability models, solar radiation estimates, habitat classification and a dozen other things. Computed on a projected raster it is a page bearing, and it inherits the same convergence.

So the aspect layer is wrong by up to twenty-six degrees at the edges of a continental sheet, systematically, in a direction that varies smoothly across the map. Downstream models take it as a compass bearing because that is what its documentation calls it.

The error is worse there than in the shading, because a tone is read qualitatively and a bearing is read as a number. A slope-aspect classification into eight sectors of forty-five degrees is shifted by more than half a sector at the sheet’s edge, so a substantial fraction of cells are placed in the wrong class by the projection alone.

The refusal #

The measurement has to be able to return zero and it does. On Mercator the declared azimuth and the ground azimuth agree to $10^{-9}$ degrees at every sampled point, and the two shadings are identical to the same precision.

That is what makes the conic numbers a measurement of convergence rather than of the shading code. A version of the check that only ever reported large numbers would be reporting a bug, and the flat case is what distinguishes the two.

What a corrected hillshade would look like #

Not very different, and that is worth saying honestly.

Correcting the azimuth per cell changes which slopes are brightest by up to a quarter of the tonal range on the worst sheets, and the picture remains a recognisable shaded relief. Nobody would look at the two side by side and call one broken.

The difference is in what a reader may conclude. On the corrected version, a slope that appears lit faces the stated compass bearing everywhere on the sheet; on the uncorrected one it faces a bearing that depends on where it is. The first supports the inference the legend invites and the second does not.

Why 315°, and what it hides #

The conventional azimuth is worth a paragraph because it explains why nobody notices.

Shading from the north-west is a convention with a perceptual reason behind it: a light from the upper left is read by most viewers as a light from above, and terrain lit from the lower right inverts — ridges read as valleys and valleys as ridges. That effect is strong, immediate and well documented, and it is why 315° is nearly universal.

It also means that the precise azimuth has never mattered much. Nobody chose 315 rather than 320 for a reason, so nobody has a stake in it being exactly 315 at every point of the sheet, and a swing of twenty-six degrees either side of it looks like a variation within an arbitrary choice.

It is not, and the reason is the aspect layer above. A tonal convention can tolerate a wandering azimuth; a derived bearing cannot, and the two come out of the same operator.

A correction that is one line #

For completeness, the correction is short enough to write down.

At each cell, compute the meridian convergence $\gamma$ of the projection at that cell’s ground position. Shade with an azimuth of $A - \gamma$ rather than $A$, where $A$ is the declared compass bearing. The result is a raster shaded as though the light came from the stated compass direction everywhere, and the legend becomes true.

The convergence has a closed form for every projection this collection carries: $n\,\Delta\lambda$ for a conic, $\arctan(\tan\Delta\lambda\sin\varphi)$ for a transverse Mercator, zero for a normal cylindrical. Computing it per cell is a handful of operations against the dozens the shading itself costs.

What it needs is for the operator to know the projection, and that is the whole obstacle. The correction is not hard; the plumbing is.

Where the model stops #

The measurements here use a Lambertian shading model — the cosine of the angle between the light and the surface normal, clamped at zero — which is what almost every implementation uses and is not what a real landscape does.

More importantly, nothing here treats the vertical exaggeration, which every published hillshade applies and none states, and which changes the apparent steepness by a factor that is usually between two and five. That factor is a much larger error than anything measured here and is a deliberate choice rather than an oversight, which is why it is not this rung’s subject.

The oblique case, which is worse #

Everything above assumes a projection in its normal aspect, where the convergence is a smooth function of longitude alone. An oblique aspect breaks even that.

The aspect is a free choice — a projection composed with a rotation of the sphere — and a rotated cylindrical or conic has a convergence that varies in two directions rather than one. The light then swings across the sheet vertically as well as horizontally, and there is no direction along which it is constant.

Oblique aspects are used for exactly the maps that are most often shaded: long diagonal mountain belts, island arcs, and any region whose long axis does not run east–west. So the case where the correction matters most is the case where the uncorrected error has the least regular structure, and a reader has the least chance of learning to allow for it.

Who found it, and when #

The convergence itself is elementary and is in every surveying text. Its application to shading appears occasionally in the cartographic literature as a note that hillshades on conic projections should have their azimuth adjusted, usually without a number attached.

The reason it stays a note is the structural one above: the correction has to be applied inside a raster operator that does not know where it is, and the tools are arranged so that it cannot be. Fixing it means passing a projection into a neighbourhood operator, which is a change to an interface rather than to an algorithm — and interfaces outlive the reasons for them.

What this rung shares with the rest of the field #

The pattern is the one this whole field is built on, arriving in a product rather than in a measurement.

A quantity is defined on the ground. It is computed on the page, because the page is where the data ended up. The result is labelled with the ground quantity’s name, and the difference between the two is the projection — which is present, computable, and absent from every part of the pipeline that could apply it.

A slope is not a shape is that argument for a magnitude. Water runs downhill on the ground is it for a direction of flow. This is it for a direction of illumination, which is the same kind of object arriving from outside the data rather than out of it.

The three together suggest the general form. Anything a raster operator computes from its own neighbours is a page quantity, and every page quantity needs the projection to become a ground one — so the question to ask of any derived layer is not whether it is correct but which frame it is correct in.

A hillshade is the sharpest instance because it is the layer a reader trusts most. Nobody measures a slope off a hillshade, but everybody reads its relief directly as terrain, and a reader who has interpreted a hillshade has formed a belief about the ground without ever having asked the data a question.

That question has an answer that is always available and almost never asked, which is what makes the class of defect so durable: nothing about a derived layer is broken, no gate could fail it, and the only way to notice is to ask which frame its numbers live in — a question a file format has nowhere to record and a pipeline has no step for.

Where the ladder goes next #

This rung is about a direction imposed on a field from outside. The next is about a quantity that is in the field — its own second derivative, which decides whether a place is a spur or a hollow, and which the page gets wrong in a way conformality does not fix.