The third point put where it looks right

The instruction is standard. Put two vanishing points on the horizon for the two horizontal directions, then put a third one a long way below the picture — or above it, for a worm’s-eye view — and run the uprights to that. How far below? Far enough that the convergence looks right.

There is nothing to judge. Given the two horizontal points and the centre of the picture, the third is determined, and there is exactly one place it can be.

Why it is determined #

Three mutually perpendicular directions in the world put their vanishing points at the corners of a triangle in the picture. The triangle has a property that does all the work here: its orthocentre is the principal point — the point where the optical axis meets the picture, which for a picture that has not been cropped is the middle of the frame.

That is a single fact and it fixes everything. Two corners and the orthocentre determine the third corner: the third lies on the perpendicular from the orthocentre to the join of the other two, at the distance that makes the orthocentre the orthocentre. One construction, no freedom.

Recovering the camera runs the same fact in the other direction — given three vanishing points found in a drawn picture, the orthocentre says where the camera was pointing and the triangle’s shape says what the focal length was.

So a three-point drawing that places its third point freely is not choosing a viewpoint. It is choosing a scene, and the scene it chooses is one where the uprights are not upright.

What the drawing depicts instead #

The right question to ask a drawing is what solid it is a picture of, and that question has a numerical answer.

Take the focal length the two horizontal points and the centre imply — that is a two-point drawing’s own commitment, and it is not in dispute. With a focal length, a vanishing point names a direction in the world. So the misplaced third point names some direction, and the question is what angle that direction makes with the two horizontal ones.

Ninety degrees is what an upright makes with the ground. Anything else is a lean.

In the book’s layout, with the horizon a little above the picture and the third point 3,354 px below the centre, misplacing that point by 260 px leans the depicted uprights by 0.58°, and by 400 px leans them 0.86°. Those are small numbers and it is worth saying so plainly: in the case a manual draws, the by-eye step is nearly harmless.

In the steeper layout — horizon 400 px above the picture’s centre, which is a phone held up at a building across the street — the third point belongs 761 px below the centre, and 400 px of misplacement leans the depicted uprights by 10.03°.

Ten degrees is not subtle. A tower drawn that way is a tower built on a slope, or a tower falling over.

Why the error hides #

The reason the same 400 px costs 0.86° in one layout and 10.03° in the other is worth spelling out, because it is the general reason this kind of mistake survives.

The third point is far from the picture in a nearly-level view and close to it in a steep one. A vanishing point a long way away names a direction close to the picture plane, and moving it a fixed distance changes that direction by an angle that falls off roughly as the square of the distance. So the further off the paper the point is, the less a hand’s-breadth of misplacement matters.

Which means the by-eye method has been calibrated, over a century of drawing manuals, on exactly the cases where it works — and every one of those cases is a case where the point being placed by eye is so far away that it is hardly being placed at all. The instruction “far enough that it looks right” is an instruction that cannot be much disobeyed when the answer is three thousand pixels down.

Photography moved the common case. A picture taken from a pavement with a wide lens has its third point on the paper, or nearly, and the method inherited from the drawing manuals is being applied where its own tolerance is ten times tighter.

The tolerance, with the constant in it #

“Falls off roughly as the square of the distance” is the right shape and it can be given a coefficient, which turns the observation into a tolerance a draughtsman could work to.

A vanishing point sitting $v$ pixels from the principal point along the vertical stands for the camera-space direction $(0, v, f)$, whose angle from the optical axis is $\arctan(v/f)$. Perturb the point by $\Delta$ and that angle — which is the lean of the depicted uprights — changes by

$$\text{lean} \approx \frac{f\,\Delta}{f^{2} + v^{2}}\ \text{radians},$$

which for $v \gg f$ is $f\Delta/v^{2}$: inverse-square in the distance, as the section above says, and linear in the focal length.

The expression can be run backwards against the essay’s own numbers, which is worth doing because neither focal length is quoted. The book’s layout has $v = 3{,}354$ px and 400 px of misplacement costing 0.86°; solving gives $f = 429$ px, which is an ordinary wide-normal lens and is the sort of camera the rest of this collection’s figures use. Checking it at the other reading: 260 px of misplacement in the same layout predicts 0.56° against the 0.58° measured, the small excess being the second-order term the linearisation drops.

Now the tolerance. Requiring the depicted uprights to lean by less than one degree means

$$\Delta < \frac{0.01745\,(f^{2} + v^{2})}{f},$$

and putting the two layouts into it gives the number this essay is really about:

In the manual’s layout the third point may be misplaced by 465 pixels — well over half the width of the picture — and in the phone’s by 31. A fifteenfold tightening, and thirty-one pixels on a sheet is about a millimetre. The first is a tolerance nobody could miss by eye and the second is one nobody could hit by eye.

Three things follow.

The method was never good; it was applied where the tolerance was enormous. A by-eye placement is accurate to perhaps a centimetre on a drawing board, which is inside 465 px and far outside 31. So the manuals’ confidence is real and it is confidence in the layout rather than in the method — a distinction that only becomes visible when the layout changes, which is what a camera pointed up at a building did.

The tolerance is a property of the picture, not of the draughtsman, so it can be computed before any drawing is made. Two horizontal vanishing points and the centre give $f$; the orthocentre condition gives $v$; and the expression gives how much slack the third point has. A manual could have printed that number beside every worked example and none does.

And the same expression says which errors are worth chasing. In a nearly level view the third point is the least sensitive quantity in the drawing, so effort spent on it is misplaced while the horizontal points — which fix the focal length and therefore the distance the picture is correct from — are worth a ruler. In a steep view the ordering reverses. That is the same pattern the conditioning of a fitted lens shows in a different subject: which parameter deserves the care is decided by the arrangement rather than by which one the method names last, and a recipe that always puts the same step last will be right about half the time.

What stays right, and why that is the problem #

The other half of the measurement is the half nothing warns about.

Whatever is done to the third point, the two horizontal directions stay exactly perpendicular to each other — to the arithmetic floor, at every misplacement tried. That is not a coincidence: the focal length was fitted from those two points and the centre, so perpendicularity between them is true by construction.

So a drawing with a badly placed third point has a perfectly square ground plan, perfectly square corners between its two horizontal directions, and uprights that lean. There is no cue anywhere in the horizontal structure. Every check a draughtsman would naturally make comes back clean.

This is the same shape of failure as the cube that is a box, where a two-point cube drawn with the two far edges placed symmetrically comes out a cube for free and eight points of asymmetry — invisible on the page — makes it a box half again as deep as it is wide. In both cases the by-eye step is exactly the step no printed method supplies a construction for, and in both cases what it decides is invisible in the drawing and enormous in the scene.

The other reading: the point is right and the centre is not #

There is a second way to interpret a drawing whose third point is off where the orthocentre condition wants it, and it is worth taking seriously because it is sometimes the correct reading.

The condition ties together three things: the two horizontal points, the third point, and the principal point. A drawing can satisfy it with the third point anywhere at all, provided the principal point moves to the orthocentre of whatever triangle results. And a principal point away from the middle of the frame is not an impossible camera — it is a cropped one, or a shifted one, which is exactly what an architectural photographer’s shift lens produces and what cropping the top off a photograph produces.

So the honest statement is a fork. A three-point drawing with its third point in the wrong place is either a picture of a scene whose uprights lean, or a picture of an upright scene taken with the optical axis somewhere other than the middle of the frame. Nothing inside the picture distinguishes them, because both readings produce the identical drawing.

That is not a defect of the analysis; it is a real ambiguity, and it is the same one what one picture determines records for the general case. A picture on its own does not know where its own optical axis was. What settles it in practice is an assumption — that the photograph was not cropped, or that the scene was built plumb — and the assumption is doing more work than the geometry.

Which reading is the charitable one depends on what produced the drawing. For a photograph, the shifted-centre reading is usually right and usually harmless: the scene is plumb and the frame has been cropped. For a drawing made by hand from a manual, the leaning-uprights reading is the right one, because nobody drawing by eye is tracking where the optical axis fell.

Placing the point deliberately #

There is a legitimate reason to move the third point, and it is not the one the manuals give.

Moving it toward the picture makes the uprights converge harder, which is what “more drama” means. But the amount of convergence is set by the tilt of the camera and the focal length, and both of those are choices a photographer or a draughtsman is entitled to make. Choosing to tilt further, or to use a shorter lens, moves the third point closer — and then the drawing is a correct picture of a scene from a correspondingly extreme viewpoint.

So the right way to get dramatic convergence is to pick the camera that produces it and let the third point follow. The wrong way is to pick the third point and let the camera be whatever it turns out to be, because what it turns out to be is usually nothing: no camera at all produces a drawing whose three points fail the orthocentre condition with the centre in the middle of the frame.

The distinction matters for the same reason it matters everywhere in this collection. A picture produced by choosing a camera is a picture of something. A picture produced by choosing where lines should meet is a picture of whatever the choices happen to be consistent with, and often that is nothing.

The construction, which takes one line #

Since the point is determined, it can be constructed, and the construction is short enough to be worth stating.

Join the two horizontal vanishing points. Drop a perpendicular from that join through the centre of the picture. The third point is on that perpendicular — and its distance from the centre is settled by the requirement that the centre be the orthocentre, which comes out as the product of the two horizontal points’ displacements from the centre, divided by the distance from the centre to their join.

One perpendicular and one length. The length is a length on the paper, computed from two other lengths on the paper, so it is a ruler operation rather than a straightedge one — which is worth noticing, because it means this construction is not in the same family as the exact ones in carrying a height across the room. It is a metric construction and it needs the principal point, which is a metric fact about the camera.

That is unavoidable. Perpendicularity is a metric relation and no amount of incidence produces it. The best a picture can do is what what one picture determines sets out: the horizon buys the affine structure, and something more — a circle, a known angle, a principal point — is needed for angles.

The level case, where there is no third point at all #

There is a degenerate case that the recipes never mention and that the machinery here refuses outright.

If the camera is level, the horizon passes through the centre of the picture. Then the join of the two horizontal vanishing points passes through the orthocentre, and the third corner of the triangle is at infinity — which is correct: for a level camera the world’s verticals are parallel in the picture and have no finite vanishing point.

So a level two-point drawing has no third point to place, and adding one is not a refinement but a claim that the camera was tilted. Books that present three-point perspective as “two-point with extra drama” have this backwards. The third point is not a stylistic addition; it is a report that the picture plane is not vertical, and the amount it is displaced says by how much.

The construction here refuses rather than returning a very large number, because a very large number would be an answer to a question that has none.

What to check in a drawing #

Three tests, in increasing order of trouble.

Is the horizon through the centre? If it is, there should be no third point, and the drawing is a two-point layout whose separation already fixes its focal length. If there is one, the drawing is claiming a tilt it has not drawn.

Is the third point on the perpendicular? Drop the perpendicular from the centre to the join of the two horizontal points and see whether the third is on it. Being off that line is not a matter of degree — it depicts a scene where the uprights are not perpendicular to either horizontal direction, which is a solid nobody meant.

Is it at the right distance along it? This is the one that can be wrong by a lot without looking wrong, and it is the one worth measuring rather than eyeballing. The check is the orthocentre condition, and it is a multiplication.

A drawing that passes all three is a picture of a real box from a real camera. One that fails only the third is a picture of a box whose uprights lean by an amount that can be computed, and computing it is the difference between a drawing that is wrong and a drawing that is wrong by 0.86°.