Dividing depth by eye

A row of fence posts recedes into the distance, a metre and a half apart. In the picture the gaps shrink. How fast?

The correct answer is a construction. What gets used instead is one of three habits, and this essay measures all three.

Equal steps. Divide the picture distance from the near post to the horizon into equal parts, and put the posts at the divisions.

Halving. Put each post halfway between the last one and the horizon.

Tapering. Reduce each gap by a fixed fraction of the last — start with a third of the way to the horizon and multiply by 0.72 each time, or whatever looks right.

Turning a drawing error into a distance #

The measurement that makes this more than a matter of opinion: a point on the ground at a given height in the picture is at exactly one depth, and that depth is recoverable.

Back-project the image point through the camera, intersect the ray with the ground plane, and read off the distance from the eye. So a row of posts placed by eye can be asked where it claims the posts are — and the answer is a list of distances in metres which are supposed to be equally spaced.

For a row spaced 1.4 m, from an eye 1.55 m up with a 44° view:

The last figure is not a typo. The tapering method’s later posts land so close to the horizon that they claim distances of tens of metres in a row that is supposed to run about eight.

Why they all fail in the same direction #

Every one of the three puts the far posts too far away, and the reason is structural.

The correct spacing is governed by a projective relation: the image height of a ground point is a fractional-linear function of its depth. A row at equal depths therefore produces image gaps that shrink toward the horizon in a very particular way — fast at first, then slowly, approaching but never reaching the horizon.

All three by-eye methods approach the horizon too fast. Equal steps reaches it in a fixed number of moves by construction. Halving reaches it geometrically. Tapering reaches it in whatever number the ratio produces. The correct construction never reaches it at all, in any finite number of posts.

That is the general shape of the error: a projective sequence being approximated by an arithmetic or a geometric one, and the two families disagree most exactly where the interesting part is.

The check that fails #

The obvious way to test a suspect row is with the invariant: take four consecutive divisions, compute their cross-ratio, and compare with the value four equally spaced points must have.

Four equally spaced points have cross-ratio 4/3. A correctly projected row of equal spacings gives 4/3, by invariance.

And a row whose divisions are equally spaced in the picture also gives 4/3 — because four equally spaced points give 4/3 wherever they are. The equal-steps method passes this test exactly, to sixteen digits.

The invariant is behaving correctly. Both configurations genuinely have the same cross-ratio, so no measurement of that cross-ratio can distinguish them. What was wrong was the inference: preserving the invariant is necessary for being a projection and is not sufficient, and a check that only knows the first half will certify a wrong construction and report perfect agreement while doing it.

The check that works #

The repair is to bring in the point the disagreement is actually about.

Use three divisions and the vanishing point as the fourth. In the world the fourth point is at infinity, so the cross-ratio collapses to a simple ratio, and for equally spaced points its value is 2.

Under that test all three methods fail: equal steps by 14%, and the other two by more. The vanishing point is what carries the information, because it is the one point whose position encodes the fact that the row is going somewhere unreachable — which is exactly the fact the by-eye methods get wrong.

There is a lesson in that beyond this subject. A test built from an invariant is only as good as the configuration it is applied to, and applying it to a configuration that both the right and the wrong answer satisfy measures nothing. The way to find out whether a check discriminates is to run it on something known to be wrong and require it to fail — which is what this site’s gate does for every assertion it makes, and which is how this one was caught.

What the correct construction costs #

Very little, which is the frustrating part.

The measuring point places equal depths exactly. It needs one extra vanishing point — the one belonging to a transfer direction — and a ruler along the ground line. Checked against the projection, it lands on the correct positions to 8 × 10⁻¹⁴ px over six divisions.

For the special case of doubling rather than arbitrary spacing there is an even cheaper construction that needs no measuring point at all: from the midpoint of the far side of a drawn rectangle, a line through its far corner meets the extended base at the far corner of the next equal rectangle. Repeated, it lays out a receding row of equal bays with nothing but a straightedge, and it is exact.

That construction has been known since the fifteenth century. It is in the manuals. It takes about as long as guessing.

Why the habits persist #

Three reasons, and none of them is that people are careless.

The error is invisible from inside the drawing. A row placed by eye recedes convincingly. Nothing about it looks wrong, because a row of posts at some set of depths is what it is a correct picture of — the depths just are not the ones intended, and that is the same condition the taught cube construction is in.

It usually does not matter. For a decorative row of trees, nobody is going to measure. The error becomes real when the drawing is meant to convey a dimension — an architectural view, a technical illustration, a reconstruction — and then it matters a great deal.

The correct construction is taught last or not at all. Depth division is the least-explained part of most treatments, and it is the part where the viewing distance enters, so leaving it out is what allows a drawing to be made without anyone deciding where the viewer stands.

What to take away #

The specific advice is short: for anything measurable, use the measuring point or the diagonal construction. Both are exact and neither is difficult.

The general point is the one this site keeps returning to. A construction that produces a plausible picture has not been checked; it has been looked at. Checking requires a second, independent route to the same picture, and where one is available the checking is cheap and the results are sometimes surprising — including, in this case, that the first check tried was itself no check at all.

What the correct spacing actually looks like #

Since the by-eye methods are all approximations to one relation, it is worth having the relation itself.

A point on the ground at depth z from the picture plane, seen by an eye at height h, images at a height above the horizon that goes as 1/(z + d), where d is the eye’s distance from the picture plane. So the image height measured down from the horizon is inversely proportional to depth.

Two consequences follow directly and both are useful without any construction.

Doubling the depth halves the distance below the horizon. A post twice as far away sits half as far below the horizon line as the near one. That is exact, it needs no measuring point, and it is the single most useful rule of thumb in the subject.

The row never reaches the horizon. Since the image height goes as 1/z and z is finite for every post, no post is ever on the horizon. The gaps shrink toward zero and the row accumulates against the horizon without touching it, which is exactly what all three by-eye methods get wrong.

The halving rule also gives a construction for the cases that matter most: to place a post at twice the distance, halve the gap to the horizon. To place one at three times, take a third. That is not a general method — it needs the depths to be in simple ratios — and it covers a great many practical cases at the cost of nothing.

Reading someone else’s drawing #

The inverse operation is worth having too, because it turns any perspective drawing into a plan.

Given the horizon and one known depth, every other ground point’s depth follows from its height below the horizon, by the inverse of the relation above. A drawing can therefore be asked what layout it depicts, and the answer is a plan that can be compared with the plan that was intended.

Applied to architectural drawings this finds a specific and common fault: rooms that are deeper or shallower than the accompanying plan says, because the perspective was drawn by eye from the plan rather than constructed from it. The two documents then describe different buildings, and it is usually the perspective that gets shown to the client.

Applied to paintings it is a tool of art history. The depths implied by a painted floor can be measured, compared with the architecture depicted, and used to say whether the painter constructed the perspective or approximated it — and, where it was constructed, what viewing distance was chosen.

The one by-eye method that is not hopeless #

For completeness: there is a fourth habit, less common than the three measured here, which is much better than any of them.

It is to place the second post by judgement and then use the diagonal construction for everything after. Draw the rectangle formed by the first two posts and the ground line; take the midpoint of its far side; the line from the near corner through that midpoint meets the extended base at the next post.

That is exact. It duplicates the first interval indefinitely, and it needs no measuring point, no ruler and no vanishing point beyond the one already in use.

So the whole error measured in this essay is avoidable by a construction that takes one extra line per post. The reason it is not used is that it must be learned, and the three that are used need no learning at all — which is a fair summary of why taught methods with free parameters survive.

Why the errors compound #

A single misplaced post is a small fault. A row of them is worse than the sum, because the errors are not independent.

Each by-eye method places each post relative to the previous one, so an early error propagates and the later posts inherit it with interest. The equal-steps method’s error grows linearly with the post number; the halving method’s grows geometrically; the tapering method’s depends on the ratio and typically runs away.

That is why the worst-case numbers in the table are so large compared with the first few intervals, which look reasonable. The first two or three posts of any of these rows are nearly right. It is the tail that fails, and the tail is the part carrying the sense of distance.

The correct constructions do not compound, because each division is placed from the ground line rather than from its neighbour. The measuring point transfers each distance independently, and the diagonal construction, although it does step from one rectangle to the next, steps exactly.

The one place the habits are defensible #

Worth ending with the case for the defence, because it exists.

For a drawing where nothing will be measured and the row is not the subject — trees along the edge of a landscape, a receding crowd, texture in a background — a by-eye spacing is fine and the labour of a construction would be wasted. Nobody will read a depth off it and no claim is being made.

The trouble is that the same habit gets carried into drawings where a claim is being made: an architectural view where the depth of the courtyard is the point, an illustration of a mechanism where the spacing of the parts matters, a reconstruction where somebody may measure. In those cases the by-eye row is asserting distances that are wrong by metres, and nothing in the drawing indicates it.

The rule that follows is about the drawing’s purpose rather than its technique: if a reader might measure it, construct it.

The floor that checks itself #

The Renaissance had a test for a depth construction that needs no arithmetic and catches every error in this essay, and it deserves to be better known.

Lay the receding row out as a floor of square tiles rather than as a line of posts. In a floor of squares, the diagonals of the tiles all lie along a few straight lines running across the whole floor, because those diagonals are parallel in the world. So the corners produced by the depth construction must be collinear along each diagonal — and if any row is misplaced, its diagonal visibly kinks at that row.

That is a construction checking itself with an incidence, which is the only thing a projection is guaranteed to preserve, and it was arrived at by people with no coordinates and no way to compute a projection at all.

Applied to the three by-eye methods it fails immediately and unmistakably: the equal-steps floor’s diagonals bow, the halving floor’s bend sharply, and the tapering floor’s fall apart entirely. The kink is visible at the second or third row, long before the errors reach the metres this essay measures.

It is also the direct ancestor of the residual reported by the least-squares vanishing point. Both ask whether things that ought to meet at a point actually do, and both treat the failure to meet as the measurement rather than as a matter of opinion about the drawing.