A slight tilt costs a construction more than a steep one

A turned cube punishes the eye, not the construction priced each mark of the measuring-point construction for a cube turned on a level camera. The method is simple: how far, in pixels of its own position on the page, can each mark be misplaced before the drawn solid is five per cent off a cube, in the worse direction? By eye, on a layout turned forty degrees, the far edge of the sliver had 0.34 px. The constructed marks kept margins of tens or hundreds of pixels, and the only one to come near being difficult was the centre of vision, at 5 px.

All of that was for a level camera. Most drawings of buildings seen from a height, or from the foot of a tower, are three-point, with the verticals meeting at a third vanishing point above or below the sheet. Three-point, laid out with a straightedge draws the layout, and nobody has priced its marks. The natural fear is a mark as expensive as the centre of vision became: a third vanishing point far off the sheet looks like a hard thing to place, and a hand drawing it has every reason to be nervous. That fear turns out to be misplaced, and the real cost sits somewhere else, at a tilt most people would regard as hardly a tilt at all.

The construction with three vanishing points #

A camera with focal length 595 px is tilted down by $\tau$, looking at the near corner of a cube turned 45° on its vertical axis. Its three edges run from the near corner to three vanishing points: two near the horizon, one far below. The triangle they make has its orthocentre at the principal point, so a person with the three points can find the centre of vision and the eye’s distance from it. That is the classical result, and recovering the camera measured how well it works.

The measuring point generalises directly. An edge running to vanishing point $V$ has its true length measured by a point $M$ on a line through $V$, at the distance $|VE|$, with $E$ the eye. In the level case the line is the horizon. In this case there are choices: $M$ can sit on the vanishing line of any plane containing the edge, so on any line through $V$. A tick on a line through the near corner, parallel to that vanishing line, marks off one side’s length on the scale of that corner. The line from the tick to $M$ cuts the edge at the far corner.

This construction lands on the cube, to under a millionth of a pixel at every tilt from 5° to 45° and at every turn tried. The check is against the full projection of the corner. A first attempt put the tick on the same side as the measuring point, and that lands on a corner at an entirely wrong place, with a clean-looking diagram: it is worth remembering when drawing one by hand that the two lie on opposite sides of their common direction.

Every mark, priced #

Take a tilt of 20° and the same five-per-cent rule, reading only the length of the edge from the near corner, since a shift of the whole drawing changes nothing about the solid. Moving a vanishing point also moves what is built from it, which is the centre of vision, the eye’s distance and the measuring point.

By eye, the far edge has 4.7 px, close to the 4.4 px the level cube had. The tick has 5.7 px, which is tight, though a tick is placed with a ruler, by measuring a length off a scale, and a ruler does this to a good deal better than 5 px on a drawing of this size. The measuring point has 27.5 px. A different vanishing point has 40.6 px, the centre of vision 59.5 px and the edge’s own vanishing point 64.1 px.

So at this tilt the ordering of the level case stands. The construction is many times more forgiving than the eye, wherever the eye’s own work has been handed over to a line. All of the difficulty the construction removes is in the far edge, and it has moved into nothing in particular.

The third point itself #

The vertical edge’s marks answer the question that was asked directly.

At 5° the third vanishing point is at row 7,011, nearly seventeen frames below the sheet. It has more than ten thousand pixels of tolerance: it could be placed wrongly by more than the distance to the sheet, and it would not matter. At 20° it stands at row 1,845 and has 761 px. At 45° it is on row 805, just off the sheet, and has 133 px. So the third vanishing point is the cheapest mark in the construction, at every tilt. The reason is that a vanishing point far away gives its lines a direction, and a direction at the near corner hardly changes when the point moves. It is exactly the same reason the measuring point stays on the sheet on a turned cube, and it answers the fear about a third mark. It does not add one as expensive as the centre became.

The vertical edge’s own by-eye price moves more than a horizontal edge’s does, from 8.8 px at 5° to 4.4 px at 45°. At 45° the vertical edge is foreshortened as much as the horizontal edges were on the original cube, and it becomes just as hard to judge. Tilt moves difficulty from the horizontal edges to the vertical one, which is the expected trade between the three directions of a cube, and the sum changes little.

Where the cost actually is #

The horizontal edges’ other marks are what holds the surprise.

By eye the far edge stays near 5 px, and the measuring point between 22 and 38 px. But a vanishing point of this edge’s, moved along with everything built from it, has 61 px at a tilt of 30°, 40.6 px at 20°, 20.7 px at 10°, 10.3 px at 5° and 6.1 px at 3°. Near level, it has become as tight as judging the far edge by eye. That reverses what a person would guess. A tilted drawing looks harder when the tilt is large, with its converging verticals and foreshortened faces, and a drawing tilted three degrees looks nearly level. On the measure that counts here, the slight tilt is the harder of the two, for a hand constructing it.

The cause is the centre of vision. In a three-point construction it is the orthocentre of the triangle. Near level, the triangle is two nearby points on a horizon and a third very far below: a long, thin triangle. An altitude from the far point is nearly vertical, and it hardly responds to anything. The altitude from a near point, on the other hand, has to be perpendicular to a long side that runs nearly vertically, so it lies nearly along the horizon, and it crosses the first at a small angle in a place that moves quickly as the near points move. So a pixel of error in one horizontal vanishing point moves the centre by many pixels, and with the centre the eye’s distance and the measuring point. The fault is not in that vanishing point’s own line. It is in how the construction uses it.

The cure is to stop finding the centre #

The test for that explanation is easy. Keep every other part of the construction and give the drawing its centre of vision from outside, as a level construction does. A person setting out a slightly tilted view generally has that anyway, since a picture’s centre of vision is usually the middle of the sheet.

With the centre given, an error in another vanishing point has no effect on the far corner at all. That is exact, not just small, and it holds because a measuring point can stand on any line through its own vanishing point. Moving the other vanishing point merely rotates which line was chosen, and the tick and the measuring point rotate together. The dashed line at the top of the figure is that case, and its tolerance runs off the scale. The edge’s own vanishing point still matters, since the edge runs to it. But the drawing’s sensitivity at small tilts has gone.

So the construction has a very simple rule, and the manuals do not put it this way. At a slight tilt, do not find the centre of vision from the triangle. Assume it. Dividing depth by eye found that a taught method can pass its own check and still mislead, and this is a milder relation of that. Two far-off lines crossing at a small angle are a poor way to find a point that a draughtsman already knows is in the middle of the sheet. For a large tilt, say past 20°, the triangle is well shaped and the orthocentre is safe to compute. Either way the third vanishing point can be placed quite loosely.

Which line the swing runs along #

That result makes a practical question arise. A measuring point can stand on any line through its vanishing point. Which is the best one? Different lines have different levers.

The natural habit, for anybody taught on a level picture, is to swing along the horizon. At a tilt of 5° that gives the measuring point 5.0 px of tolerance, little better than the eye, against 22.6 px when it swings along the upright line through the third vanishing point. At 10°, 10.0 px against 24.1 px. At 20°, 19.3 px against 27.5 px. At 30°, 28.7 px against 31.5 px. So the choice matters most at small tilts, by more than four to one at 5°. The tick also has a tolerance of 1.1 px on the horizon route at 5°, against 5.0 px on the upright one. That is a very substantial change for what looks like a choice between two diagrams.

What decides it is the lever. The tick-to-measuring-point line cuts the edge at an angle, and a shallow crossing is what makes a mark expensive. Swinging along the horizon at small tilts puts the measuring point nearly in line with the edge, so that the crossing is shallow. A swing along the upright line puts it far off to one side, and the crossing is clean. So the rule is the obvious one for any intersection: choose the line that crosses at the larger angle. It is no surprise on the page, and it is rarely in the instructions.

Turned as well as tilted #

The earlier essay’s finding was that the turn punishes the sliver, by eye. Its effect survives the tilt.

At a tilt of 20°, turned 40°, the sliver’s far edge by eye has 1.9 px, and the wide side’s has 8.8 px. The construction’s margins also move, in the same sense. The wide side’s measuring point rises to 352 px. But the sliver’s falls from 27.5 px to 2.5 px, and its tick to 0.7 px. On this tilted picture, then, the construction’s margin in the sliver has gone. Earlier, for a level cube, the same turn left the sliver’s measuring point at 29.9 px. A tilt combined with a strong turn is the first arrangement in this measurement where a constructed mark comes close to the by-eye one, and the advice to swing on a different line for that edge, or to avoid so strong a turn, is what’s left.

The far edges at that turn are only a few pixels from the near edge in places, so the five-per-cent criterion is itself very demanding, and a hand could not read its own work any better than that. So this should be read as the construction running out of room on a sliver. It should not be read as a fault in the method, for a picture that is not going to resolve that face anyway. A square plan is not a cube had its first useful price from a much gentler layout, and by the time the layout is this extreme, no way of drawing it is going to feel comfortable.

What this says about tilted views #

Three plain statements come out.

First, the instrument a beginner fears most, the far-off vanishing point, is nothing to fear. Its position barely enters. A large drawing board’s practical trick of running the verticals to a point on a distant wall, or of using a template of convergent lines, is more than precise enough, by a factor of thousands at small tilts.

Second, a slight tilt is more demanding than a steep one if the construction is run by the book. This is not a matter of skill. It is a conditioning, and it goes away with a one-line change to the method. So the classical method for three-point, which treats the third vanishing point as equal in standing with the others and derives the centre from all three, is at its most fragile in the very case where the third point adds least.

Third, the eye does not change. Its tolerance stays near 5 px at every tilt for the horizontal edges, and it never had anything like the margins the construction provides. Nothing about tilting a picture has made freehand placement harder than it was, by a measurable amount, except on the vertical edge at steep tilts. That was already hard, and a tilt merely moves it around between the edges.

The sixty-degree cone of vision is one more taught rule of this kind, and the same habit of mind applies to all such rules. When a method is presented as a sequence of steps, each step’s precision is assumed. Pricing them separately often finds that one step, here a step that nobody considered a measurement at all, carries most of the risk.

Tolerances stated one mark at a time #

The camera has a known focal length in the sense that its marks are computed, and the near corner is at the frame’s centre. Off-centre near corners change the triangle’s shape and could move each number somewhat. The “worse direction” of a slip is tried at sixteen angles rather than being solved. Sharp minima between those would be missed, but the margins here are large enough that this does not change any ordering.

A hand’s errors are not independent. Slips in a vanishing point and in the line to it are correlated, and a person drawing to a far-off point may do so with a ruled guide, which gives a very different kind of error from a mark placed on the paper. All the tolerances are stated one mark at a time, as they were for the level case.

And the five-per-cent criterion treats all three edges equally. A drawing where one edge has been foreshortened to a sliver might tolerate a much larger proportional error there, since a viewer’s own judgement of that edge is poor. The criterion is about the solid. It is not a statement about what can be seen.

Still open: a drawing with no third point on it at all #

The third vanishing point’s tolerance suggests a question. Since it matters so little at small tilts, a draughtsman might not need to place it. Verticals could be drawn converging by a stated small angle, picked from a table of tilts, with no third point marked at all. Each vertical’s direction would then be set independently, not by a common point, and that is the kind of method a hand can do in a moment.

The measurement that settles what that costs takes a tilted cube and a row of buildings, draws each vertical by a direction computed from its column and a stated tilt rather than from a point, adds a hand’s error of a fraction of a degree to each, and compares the solid’s error, and how well the verticals still agree on a point, with the construction priced above. It asks at what tilt the shortcut starts to fail; whether a reader could detect the failure, since verticals that do not meet at one point are the very thing both vanishing points on the paper would look for; and whether the shortcut is what many fluent drawings have really been doing all along.

A construction’s fragile step need not be the one that looks hard. Here the far-off point is cheap and the point in the middle of the sheet is expensive, unless the draughtsman simply says where it is.