A shadow decides which landscape it is

A seam breaks direction, not size sorted what happens to an object crossing the join between two bands of a several-station landscape. A road’s drawn width and a post’s drawn height come through the join to the last digit, because the eye’s height cancels out of any size taken at one depth. Where those sizes sit does not come through: the road’s edges turn 23.2° from each other and the post’s foot lands 64.6 px out of place, and a painter butting two bands can absorb the offset and can never absorb the turn.

One kind of receding feature was left out of that account, and it is the one whose own geometry is a projection. A shadow cast across the ground by a low sun runs in depth, so it crosses a join like a road. But a shadow’s direction on the page is not set by the thing that casts it. It is set by the sun’s own vanishing point, and a vanishing point is a property of the camera that drew it.

So the question is whether two bands drawn from two stations share the sun. If they do not, a picture of a landscape holds two lights, which is a much louder failure than a road that bends, and it would put a second constraint on where a join can go. They do — exactly, and for a reason worth having. And having got that far, the shadow turns out to settle something else this field had given up on.

The sun is shared, and it could not have been otherwise #

The three bands of the convention are drawn by three level cameras of one focal length, at heights of 1.6, 4.0 and 11.0 metres. They differ in where they are and in nothing else.

A vanishing point is the image of a direction, and the image of a direction depends on the camera’s orientation and its focal length and on nothing else — moving a camera without turning it moves every point in the picture and leaves every vanishing point exactly where it was. The sun is a direction. Its shadows run in another direction, the horizontal part of the first. So both of the sun’s page points are identical in all three bands, to the arithmetic floor and not merely to a tolerance.

The consequence is a reader’s reading. The sun’s altitude recovered from a band is an angle between two directions, read off those two vanishing points and the focal length, with the camera’s height appearing nowhere in it. All three bands report 22.000000 degrees. A reader fitting a single light to the whole picture finds one, and the shadows are no evidence at all about how many stations the painter used.

It is worth being clear about how little that costs the convention and how much it could have cost. Three distances in one landscape found the furthest band given 6.9 times the picture one camera would allow it, bought with a jump in the rate at which depth runs — 2.50 at the first join and 2.75 at the second. A convention that gained that much room and also had to explain two suns in one sky would be in trouble that mist could not cover, because a reader who can see both shadows can compare them without knowing anything about depth. The trade the convention makes is confined to positions, and a sun is not one.

That answers the question a seam breaks direction, not size left, and answers it in the quiet direction. It is also the second time this field has found a quantity that survives a seam because it is a fact about direction rather than about position: a landscape that changes its rule halfway up found each band handing back its own station height and no band able to hand back a common ground, and the horizon — which is where these two vanishing points live — is the one line all three bands share.

A shadow is not a direction #

Sharing a vanishing point is not the same as being drawn alike. A shadow is a segment: it runs from a post’s foot to a tip, both of which are particular points of the world, and particular points are exactly what a change of station moves.

The foot jumps 84.0 pixels — the riser’s own height at that depth, exactly as the post’s foot did. The tip jumps 51.1 pixels, and it jumps less because it is further away and the same difference of eye heights buys fewer pixels there. The two jumps are unequal, and that is the whole of the new result, because the difference between them is the shadow’s length.

Drawn by the near band the shadow is 159.7 pixels long; drawn by the far band, 167.4 — a ratio of 1.0483. A road’s drawn width does not change across a join and a post’s drawn height does not change across a join, because each is a size taken at one depth and the eye’s height cancels out of it. A shadow’s length is a size taken across two depths, and nothing cancels: it is the one drawn size in this landscape that a seam is free to alter.

So that split holds and gains a third category. Sizes at one depth survive. Directions survive, when they are directions and not the drawn slopes of particular lines. Lengths in depth do not.

No place for the sun is safe #

If the seam turns a shadow, a painter has an obvious defence: put the sun where the turn is smallest. The defence does not exist, and the reason it does not is a trade.

With the sun straight behind the viewer the shadow runs down the picture’s own column. A change of eye height moves that shadow without turning it, so the join leaves no angle at all — and it draws the shadow 2.50 times as long under the far band as under the near, because the whole of the dislocation has gone into the length. Turn the sun a little and the length error falls away fast while the turn climbs, reaching 25.4° at an azimuth of seven degrees. Turn it further and both subside: by sixty-five degrees the turn is 3.2° and the length ratio 1.004.

Two things follow. The painter’s best place for the sun is well round to the side, which is not where a landscape painter usually puts it, and the worst is a few degrees off dead behind — close enough to the obvious choice to be the likely one.

And the two symptoms are never both absent, because they are one dislocation resolved two ways. The far band draws the shadow’s foot 84.0 px lower than the near band does and its tip 51.1 px lower; the difference between those two numbers is a vector, and where it points relative to the shadow decides which symptom a reader sees. A shadow running down the page has that difference along itself, which is a change of length and no turn. A shadow running across the page has it almost perpendicular, which is a turn and almost no change of length. Nothing can make the difference vanish, because it is the riser’s image at two depths and the two depths are not the same.

The shadow’s foot moves 84.0 pixels at every azimuth, because where the foot is drawn depends on its depth and its depth does not depend on where the sun is. That is the part a painter can absorb, and it is the same offset a seam breaks direction, not size found a painter absorbing by choosing where to butt the bands.

The reading a shadow decides #

The stations are also a staircase found something this field has been carrying ever since. A level eye draws a ground point on a row that depends on the eye’s height and the ground’s only through their difference, so a landscape drawn from three stations over flat ground is, to $6\times10^{-14}$ pixels across ninety-nine samples, the same picture as one eye at eleven metres over ground stepped 9.4 m and 7.0 m above the far plain. Two readings, one picture, and nothing in the marks to choose between them — the discriminating evidence would be the riser itself, a cliff 103% of a band’s own height, which is exactly where the convention lays its mist.

A shadow is not a ground point, and that exemption is the whole of what follows.

Under the flat reading the ray from the post’s top descends at the sun’s altitude and meets the ground 14.9 m from the post, at 34.85 m. Under the terraced reading the ground beyond the join is 2.4 m lower, so the ray must descend 2.4 m further before it lands, which takes another 2.4 m divided by the tangent of the altitude — 5.94 m of horizontal run. The shadow ends at 40.79 m.

Those are different depths, so they are drawn on different rows: 11.7 pixels apart, in a picture whose every ground point the two readings place identically to fourteen decimal places. One shadow crossing one join says which landscape a reader is looking at.

The mechanism is worth stating plainly, because it is not that the shadow is drawn by different rules. It is that the shadow is a consequence of the ground rather than a part of it. The two readings agree about where every piece of ground is drawn and disagree about the shape of the ground in the depicted world, and a ray of light is sensitive to shape where a projection of the surface is not.

Whatever the hour #

A discrimination that worked only at one sun would be a curiosity. This one does not depend on the hour in any useful sense.

A lower sun throws the extra run further: 13.6 m at ten degrees against 2.9 m at forty. But it throws it into deeper parts of the picture, where a metre of depth is drawn with fewer pixels. The two effects very nearly cancel, and the gap stays between 9.8 and 11.7 pixels across every sun that lays a shadow across the join at all.

That flatness is what makes the test usable on a real painting, where nobody knows the altitude. A reader who measures the drawn tip of a shadow crossing a seam, and computes where each reading would put it, is comparing numbers about ten pixels apart whatever they assume about the sun — and the assumption they have to make is the one the reading itself supplies, since the sun’s altitude is recovered from the same picture at 22.000000 degrees.

And it measures the riser it found #

A test that says which of two readings a picture is has done its work. This one does more, because the quantity it compares is not a yes or a no but a length, and a length can be inverted.

The extra run a crossing shadow makes under the terraced reading is the riser divided by the tangent of the sun’s altitude. The altitude is known — it is read off the two vanishing points at 22.000000 degrees, from the same picture, with no station height in it. So a reader who measures where a crossing shadow actually ends recovers the riser directly: multiply the shadow’s excess run by the tangent of the altitude and the answer is the step in the depicted ground, which is also the difference between the two stations the painter used.

What that recovery is worth depends on how well the tip can be read, and the sensitivity is nearly the same flat curve as before. A pixel of error in the drawn tip costs 0.252 m of riser at a sun of fourteen degrees, 0.237 m at twenty-two and 0.251 m at thirty-four — a quarter of a metre a pixel, whatever the hour, for the same reason the discrimination held: a low sun puts the tip further away where a pixel is worth more metres, and the longer lever divides the extra metres back out again.

Against a riser of 2.4 m that is a tenth of the quantity for each pixel of reading error, so a tip placed to two or three pixels gives the step between two stations to within about a quarter of it. A landscape that changes its rule halfway up recovered each band’s own station height exactly and could not recover a common ground across a join; this recovers the thing that separates them, from a single shadow, and it does so without needing the cliff that reading said would be required.

The two recoveries are not the same measurement twice. One reads each band’s marks against the assumption that the band’s ground is flat. The other reads one shadow against the assumption that light travels in a straight line, and the second assumption is the one a painter is least likely to have violated deliberately.

What this does not settle #

The riser is the convention’s, not the painting’s. The 2.4 m step is the difference between two station heights this convention uses. A real landscape of this kind may be drawn from stations nobody has measured, and the discrimination is between two readings of a stated pair of stations rather than a recovery of them.

A painter need not draw a shadow that crosses. Everything above requires a shadow that reaches past the join, which needs a tall enough object near enough to the seam under a low enough sun. A painting whose seams fall in empty ground has no crossing shadow for the same reason it has no crossing road, and the stations are also a staircase already suspected the choice of seam depth is not free.

A cast shadow is not always a cast shadow. Painters of this convention draw shading, foliage and reflected darkness that are not projections of anything, and treating a dark shape as the image of a ray is a judgement about the picture before it is a measurement of it. The reading applies to a shadow that is being drawn as a shadow, and it has no way to tell one from a dark rock.

And the sun was a point at infinity throughout. A light at a finite distance has vanishing points that are not shared between two stations, because the direction from an object to the light then depends on where the object is. That case is a different measurement and would break the one clean result above — the sun is shared exactly because it is infinitely far away.

Three results about one light #

Every band of a several-station landscape images the sun’s direction, and the ground direction its shadows run in, at the same page point to the arithmetic floor, because a vanishing point is a fact about where a camera points and these stations differ only in where a camera is. Each band reports the altitude as 22.000000 degrees. A picture’s shadows convict nobody of a second viewpoint.

They are dislocated at the join all the same, and they break a rule a seam breaks direction, not size had established: a shadow’s drawn length changes across a seam — 159.7 pixels to 167.4 at one sun, and by a factor of 2.50 with the sun straight behind the viewer — where a road’s width and a post’s height do not, because a shadow is a length taken across two depths. There is no azimuth that avoids the seam; there is only a choice between a turn of up to 25.4° and a length error of up to two and a half times.

And a shadow decides what no ground point could. The two readings of this convention — three stations over flat ground, or one station over a staircase — agree about every drawn ground point to fourteen decimal places and put a crossing shadow’s tip 5.94 m apart in the world and 11.7 px apart on the page, at a gap that stays between 9.8 and 11.7 px for every sun that casts across the join.

Still open: what a shadow says about where the seam is #

The measurement above assumes the join’s depth is known, because it is asking which of two readings a picture is rather than reading anything out of it. A reader of a real painting does not know where the seam falls; finding it is the first problem, and the mist is laid precisely to make it hard.

A crossing shadow may solve that problem rather than depend on its solution. A shadow drawn in one continuous stroke by a painter working in a single band is a straight line on the page; one that crosses a seam is two segments meeting at an angle, and the angle’s vertex is at the join depth. So the seam’s depth is written into the picture at the kink of every shadow that crosses it, at a place the mist cannot cover without covering the shadow too.

The measurement that follows takes a landscape whose join depth is hidden, lays several shadows across it at different azimuths and altitudes, and asks how precisely the vertex of each locates the seam: how far a reader’s estimate of the join depth is out for a shadow read to a pixel, whether shadows at different azimuths agree, and how few are needed before the answer is better than what the bands’ own rate change gives. If a shadow locates a seam more sharply than the discontinuity in the ground does, then the convention’s mist has been hiding the wrong thing.