A turned cube punishes the eye, not the construction

A square plan is not a cube finished the two-point cube with its measuring points and priced every mark a hand has to make. On the layout the site’s cube figures share — vanishing points 1,190 px apart, the cube’s two faces turned equally from the picture — a far edge placed by eye had to be within 4.4 px of the right place before the drawn solid was five per cent off a cube, while a measuring point could slip 42.1 px and the centre of vision 29.0. The construction forgives its marks about ten times as much as the eye is forgiven, through a lever: each constructed corner moves about a ninth of a pixel for a pixel of slip.

Every one of those numbers came from a symmetric layout. The essay ended by turning the cube in its mind: rotate it so one face is nearly square to the picture, and one vanishing point runs far off the sheet. It expected the measuring point to run with its vanishing point and its lever to lengthen, and asked whether the construction’s advantage would collapse on that side — whether a measuring point hundreds of pixels off the sheet would still forgive its slip by a factor of ten, or whether at some turn the taught shortcut and the finished construction would come to cost the same.

The construction’s advantage does not collapse. The expectation that it would rested on the measuring point running away, and it does not.

Turning the cube without changing the lens #

The measurement turns the cube and holds everything else. The focal length stays at 595 px, the principal point at the middle of the frame, and the near vertical edge at the same place and the same 202 px height. What moves is the two vanishing points: turning the cube by an angle moves the right one out along the horizon as the tangent of the turn and the left one in as its cotangent, so their distances from the principal point still multiply to the focal length squared. Every layout is a picture of a cube through the same lens.

Turned thirty degrees, the right face is nearly square to the picture and the left face has become a steep sliver. The left vanishing point has come in to 186 px across, close to the near edge at 330; the right one has run out to 2,566 px, nearly four sheets to the right. The construction still lands on a cube, to 3 × 10⁻¹³, and the two far edges now fall at different fractions of the way to their vanishing points — 24.7 and 8.1 per cent. Symmetry is no longer the target, which the cube that is a box warned it would stop being.

And the two measuring points land at 802 and 267 px. Neither has gone anywhere near 2,566.

The measuring point stays on the sheet #

That is the fact the earlier expectation missed, and it is worth seeing across the whole turn.

As the turn goes from nothing to forty degrees, the runaway vanishing point goes from 940 px to 7,146 px — ten sheets off. Its measuring point, swung from it through the station point, goes from 99 to 319 px and never leaves the sheet. The other vanishing point comes in from −250 to 293 px, and its measuring point goes out from 591 to 890, just past the sheet’s right edge.

The reason is in what a measuring point is. It is the vanishing point’s distance from the eye, laid back along the horizon from the vanishing point toward the middle. When a vanishing point is near the principal point, that distance is mostly the focal length and the measuring point lands well away from it. When a vanishing point runs off to thousands of pixels, its distance from the eye is almost the same as its distance along the horizon, so laying it back from the vanishing point brings the measuring point almost all the way home — to within the focal length’s contribution of the principal point. The further the vanishing point runs, the nearer the middle its measuring point lands.

So the construction’s awkwardness is not where the essay feared. The vanishing point is off the sheet, and a drawer working on paper cannot mark it; but the drawer does not need to mark it to draw the receding edges if a template or a second vanishing-point rig is at hand, and the measuring point the construction actually uses to place the corner sits on the paper.

Every mark, priced again #

The prices are measured the same way as before: how far, in pixels of its own position on the page, each mark may be misplaced before the drawn solid is five per cent off a cube in either its plan or its height, in the worse direction. By eye there are now two different marks, since the two far edges are no longer mirror images: the far edge of the sliver and the far edge of the face turning square to the picture. Constructed, there are the two measuring points and the centre of vision.

The by-eye marks go in opposite directions. The far edge of the face turning square to the picture gets easier, 4.7 px at no turn to 9.5 at forty degrees, because a face nearly square to the picture is nearly drawn at full size and a pixel of misplacement is a small fraction of it. The far edge of the sliver gets very much harder: 4.4 px at no turn, 0.34 px at forty degrees. The sliver’s whole receding edge is only 37 px long by then, and its far corner sits nine pixels from the near edge; a third of a pixel is the whole of the tolerance.

The constructed marks mostly hold. The measuring point on the sliver’s side goes from 42.1 to 29.9 px. The measuring point on the wide face’s side goes from 40.1 to 325 px — it forgives more the further the cube is turned. And the centre of vision, which was the construction’s least forgiving mark to begin with, goes from 29.0 to 5.0 px. It is the one constructed mark that becomes expensive.

Where the eye’s target goes #

It helps to see why the sliver’s far edge becomes so hard to place, because the reason is not that the sliver is small. It is that the right place for its far edge stops being anywhere an eye would look for it.

At the even-handed layout both far edges belong 19.4 per cent of the way to their vanishing points, and a drawer who places them symmetrically at least gets the plan square. Turned, the two fractions part: at forty degrees the sliver’s far edge belongs 25.3 per cent of the way along a receding edge 37 px long, and the wide face’s far edge 2.9 per cent of the way along one that is thousands of pixels long. The first is nine pixels from the near edge; the second is two hundred. Neither number is one a drawer carries in the head, and there is no symmetry left to fall back on.

So the eye is being asked for two unrelated placements on two very different scales, and it gets the large one roughly right and the small one wrong by more than its whole tolerance. One, two and three point is the reminder that the choice among two-point layouts is a choice of how the object is turned, and that every turn is a different drawing problem; the taught shortcut is really a solution to one of them, the even-handed one, and the one where the cube’s faces are least interesting.

The measuring point, by contrast, is placed by the same compass swing at every turn. The distance point is the viewing distance found that the one-point version’s distance point encodes where the eye stands; the measuring points do the same for two directions, and that is why they carry the turn without the drawer having to.

The advantage grows rather than falling to one #

The essay’s question was at what turn, if any, the construction’s advantage over the eye falls to one. It can be read straight off the table.

On the sliver’s side, the measuring point against the by-eye far edge forgives 9.7 times as much at no turn, 14.4 at twenty degrees and 88.5 at forty. On the wide face’s side, 8.5, 9.5 and 34.2. Taking the least forgiving mark of each kind — the centre of vision on one side, the sliver’s far edge on the other — the ratio is 6.7 at no turn, 9.9 at twenty and 14.9 at forty.

So the ratio never approaches one. It grows on both sides and overall, and the reason is that the by-eye placement degrades faster than anything the construction asks for. The shortcut was always asking a hand to find, unaided, a point whose position depends on the lens and the turn; turned toward the picture, that point sits in a sliver of the drawing where it has to be found to a third of a pixel. The construction finds the same point through a line from a mark on the paper, and the line does not care that the sliver is thin.

The lever shortens on both sides #

The lever that made the measuring point forgiving at the even-handed layout is still there, and it shortens as the cube turns — the opposite of the lengthening the earlier essay expected.

On the sliver’s side the corner moves 0.110 of a pixel per pixel of measuring-point slip at no turn and 0.040 at forty degrees; on the wide face’s side, 0.115 and 0.028. The corner lies on the measuring line near its ground mark, and as the cube turns the corner lies closer to the ground-mark end of its line, so a slip at the far end swings it less.

The lever does not settle the tolerance alone. The sliver’s corner has a shorter lever and still loses some tolerance — 42.1 to 29.9 px — because the sliver’s shape is so sensitive to where that corner lands that a smaller movement matters more. The wide face’s corner has a shorter lever and a shape that barely cares, which is why its measuring point ends up forgiving 325 px. The two effects multiply, and on the wide side they agree.

The centre of vision is the mark to watch #

That leaves the centre of vision, which falls from 29.0 to 5.0 px and is the only constructed mark that approaches the difficulty of a by-eye placement at even-handed layouts.

One way to see why: moving the centre of vision along the horizon slides the station point round the half-circle through the two vanishing points, and moves both measuring points. At the even-handed layout it moves them the same way by about the same amount, which is the kind of slip — an asymmetry between the two sides — that the square-plan test catches. Turned, the half-circle’s diameter is long and lopsided: the station point sits on it far from its middle, where the circle is steep, and a small slide of the foot moves the station point and both measuring points further. That is where the construction’s weakness moves to, and it is a weakness a drawer can guard against, because the centre of vision is the one mark on the board that can be checked independently — it is where the picture’s principal point is, which the principal point is not the centre is about for a photograph and which for a drawing is the drawer’s own choice.

What a drawer should take from it #

Three things, in order of use.

A turned cube is where the construction pays most. The even-handed cube is the case in which a practised eye can get away with the shortcut — four and a half pixels is a line’s width, and a steady hand holds it. Turned so one face is nearly frontal, the other face’s far edge must be placed to a third of a pixel, which nobody does by eye. The construction is worth the most exactly where drawings of buildings most often put a cube: one face toward the viewer.

The far vanishing point is the construction’s problem, not its measuring point’s. Drawing the receding edges of the nearly frontal face needs their vanishing point, and at forty degrees that is ten sheets away; dividing to a point off the board is about the ways a drawer gets lines to a point that is not on the paper. The measuring point for the same side sits on the paper.

Guard the centre of vision. It is the only mark whose tolerance falls far as the cube turns, to five pixels at forty degrees. It is also the mark a drawer can place by a rule rather than by eye, so the cost is avoidable.

Where this reading stops #

One lens and one near edge. Every turned layout keeps the focal length of 595 px and the near edge at the same place and height. A longer lens puts the vanishing points further out at every turn and would change the numbers; the shape of the comparison — the sliver’s by-eye edge collapsing, the constructed marks holding — is what carries over.

Turns up to forty degrees. Past forty-five degrees the left face would become the nearly frontal one and the roles would swap; at exactly forty-five one vanishing point is at infinity and the construction as written has no measuring point on that side. The sweep stops at forty.

Five per cent is a stated limit. The tolerances are for a solid five per cent off a cube in plan or height. A stricter limit shrinks every tolerance roughly in proportion; the ratios between them are what the essay rests on.

And the vanishing points are taken as exact. As in the essay before it, the tolerances hold the vanishing points fixed and ask about the marks drawn from them. A drawer who cannot reach the far vanishing point and approximates its direction has a different error, not priced here; dividing depth by eye is about what happens when a receding edge is drawn by feel.

Turned, and priced #

Turning a two-point cube toward the picture sends one vanishing point off the sheet — 2,566 px at thirty degrees, 7,146 at forty — but not its measuring point, which is that vanishing point’s distance from the eye laid back toward the middle and lands at 319 px at forty degrees. The construction depicts a cube at every turn, with its two far edges at different fractions of the way to their vanishing points.

Priced mark by mark, the by-eye placement of the far edge of the face that becomes a sliver collapses from 4.4 px to 0.34, while the by-eye edge of the face turning square gets easier, 4.7 to 9.5. The constructed marks hold — the sliver’s measuring point 42.1 to 29.9, the wide face’s 40.1 to 325 — except the centre of vision, 29.0 to 5.0. The construction’s advantage over the eye rises from 6.7 to 14.9 for the weakest marks and to 88.5 on the sliver’s side. It never falls toward one.

Still open: whether a cube turned and tilted keeps the lever #

Everything here keeps the camera level, so the cube’s vertical edges stay vertical and parallel on the page and the construction needs only two vanishing points. Most drawings of buildings seen from a height, or from the foot of a tower, are three-point: the verticals converge to a third vanishing point above or below the picture.

The measuring-point construction has a three-point version, with a measuring point for each of the three directions swung from a station point that is no longer on the horizon’s perpendicular through a single centre. Three-point, laid out with a straightedge draws its layout; nobody has priced its marks. The measurement that follows tilts the camera by a stated angle, builds the three-point construction for the same cube, and prices every mark against the by-eye placement of the far edges — asking whether the tilt, like the turn, leaves the construction’s marks forgiving and moves the difficulty onto the eye, or whether a third vanishing point adds a mark as expensive as the centre of vision became here.