The horizon, and the fraction

The most useful sentence in perspective drawing is that the horizon is at eye level. It is the reason a photograph says how tall the photographer was, the reason a figure standing further away is cut by the horizon at the same place on their body, and the reason an illustrator can place a hundred figures consistently with one ruled line.

The rung below states its condition in its title: if the picture plane is vertical. This essay is what happens when it is not — and the answer is that the sentence splits into two sentences, one of which survives untouched and one of which does not survive at all.

Two statements that are usually one #

The incidence. A world point at exactly the camera’s eye height images on the horizon.

The ratio. The horizon cuts an upright’s drawn image at the fraction eye-height-over-object-height of its drawn length.

With a vertical picture plane both are true and nobody separates them. Tilt the plane and the first is still true, exactly, and the second is not.

The first survives because it is about which ray a point is on. The line from the eye to a point at eye height is horizontal; a horizontal line’s vanishing point lies on the horizon; and a point’s image is where its own ray pierces the picture, which is where that ray’s direction vanishes. So the point images on the horizon whatever surface the picture is cast onto, because the ray does not care what plane it is being intersected with.

The second fails because it is about a ratio along a line, and a ratio along a line is exactly what a projection destroys. With the plane vertical, the world’s verticals are parallel to the plane, their vanishing point is at infinity, and the map from world height to image height along an upright is affine — so ratios survive. Tilt the plane and that vanishing point becomes finite, the map becomes projective, and the fraction moves.

How large the failure is #

Worth quantifying, because the answer is “small, and exactly zero only at zero”, which is the shape that gets rules of thumb into trouble.

At 4° of tilt the four fractions spread by 0.06 percentage points, and at 6° they spread by 0.08. On a picture 430 px tall, a tenth of a percentage point of a 200 px figure is a fifth of a pixel.

So as a drawing rule it is safe at any tilt an illustrator would use without noticing. As a measurement rule it is not a rule at all: it holds exactly at one value of a parameter and approximately elsewhere, and the approximation is a systematic bias rather than a random error — every figure is displaced the same way, so averaging over figures does not help.

That distinction is this site’s recurring one. A construction that is nearly right everywhere is a fine drawing aid and a bad measurement, and the way to tell them apart is to ask whether the error goes to zero as anything is refined. Here it does not: it goes to zero as the tilt goes to zero, and the tilt is a property of the photograph rather than of the care taken with it.

Why the affine case is the one everybody learned #

There is a reason the ratio version is the one in every drawing manual, and it is not carelessness.

A vertical picture plane is what a camera on a tripod produces, what a draughtsman working on an upright board produces, and what a view camera with its back plumbed produces. It is the default in the strong sense that leaving it takes an action — tilting the camera — rather than an omission.

And within that default the vertical direction is parallel to the picture plane, which makes the map from world heights to drawn heights affine, which makes every ratio along a vertical survive. The manual’s rule is not an approximation to the cross-ratio; it is the cross-ratio evaluated where one of its four points has gone to infinity, which is the same relationship a parallel projection has to a perspective one.

So the two versions of the rule sit in the same relation as the two families of projection this whole site is organised around. One is exact under a condition and simpler; the other is exact always and needs one more point. Neither is the correction of the other.

The three quantities move together #

The tilt does three things at once, and it is worth seeing them as one thing because a reader who notices any of the three has noticed the other two.

The verticals converge — 3.59° at 14° of tilt, 6.50° at 26°.

The horizon leaves the frame’s middle — 213 px and 416 px.

And the vertical vanishing point arrives from infinity — 3425 px and 1751 px from the principal point.

Their product is the focal length squared, at every tilt: the pole-and-polar identity. Which means a picture that betrays one of them betrays all three, and any of them can be used to find the fourth point the height measurement needs.

Practically, the convergence is the one a reader sees first and the vanishing point is the one the arithmetic wants — and getting from the first to the second is a straightedge and an intersection.

The repair is the fourth point #

The measurement the fraction was standing in for is available exactly, and the way to get it is the site’s standard construction: replace the ratio with a cross-ratio.

A ratio along a line needs three points and is destroyed by projection. A cross-ratio needs four and survives. The fourth point on an upright’s line is the vertical vanishing point — which, when the plane is vertical, is at infinity and quietly turns the cross-ratio back into the ratio.

So the general statement, valid at any tilt:

The base, the horizon crossing, the top and the vertical vanishing point have a cross-ratio, and it gives the height.

That is the single-view height measurement, and this is why it is written with four points rather than three. On a level photograph the fourth point is at infinity and can be left out, which is exactly the shortcut the fraction rule is.

Where the horizon actually is #

If the fraction cannot be trusted on a tilted picture, the horizon has to be found some other way, and the honest answer is that it always did.

The horizon is the vanishing line of the ground plane — the image of the ground’s points at infinity. It is found where images of parallel horizontal lines meet, and that construction is independent of the picture plane’s orientation.

What is not the horizon:

The middle of the frame. True only for a vertical plane and a centred principal point. On a picture taken pointing up, the horizon drops by $f\tan\theta$ — 213 px at 14° on the camera in these figures.

The line where the ground meets the sky. That is a distant object, and on any real terrain it is not the vanishing line: hills sit above it and the sea’s edge sits below it by the amount the Earth curves.

The apparent middle of a row of figures. Which is the fraction rule used backwards, and inherits its condition.

What the sentence is worth after all this #

The rule is not being demolished, and it is worth stating what remains, because what remains is most of it.

The incidence is exact and unconditional. A point at the camera’s own eye height is on the horizon in the picture, whatever the camera was doing. So a photograph does say the photographer’s eye height, as long as there is something in it of known height standing on the ground — the horizon crosses it, and the crossing point is at eye height on that object, in the world.

The ratio is exact when the plane is vertical, which covers most photographs taken without pointing the camera up or down, and every photograph made with a shift lens or corrected afterwards.

And the cross-ratio is exact always, at the cost of one more point.

The ladder is the same three-step shape the foundations field is built on: a ratio that is destroyed, an invariant that survives, and the recognition that the familiar rule is the invariant with one point at infinity.

The horizon is not in the picture #

A last clarification, because the word does two jobs and the confusion is common enough to be worth separating explicitly.

The horizon as a line of the picture is the image of the ground plane’s points at infinity. It is a construction, found from parallels, and it exists whether or not anything in the scene is anywhere near it. In a photograph taken indoors it is a perfectly definite line crossing the furniture.

The horizon as a thing in the world is where the ground stops being visible, which on a real planet is a few kilometres away and is below the geometric horizon by the amount the surface has curved. On land it is wherever the terrain gets in the way, which is not a straight line and is not at eye level.

The rule is about the first. A figure standing at the far end of a room is cut by the geometric horizon at their eye height as reliably as one standing in a field, and the room’s walls have nothing to do with it.

This is also why the horizon can be off the frame entirely — pointing the camera far enough up puts it below the bottom edge — and the construction still works. A line does not need to be visible to be used; it needs to be findable, and two pairs of parallels find it.

The plan view, where both statements go #

The extreme case is worth a paragraph because it says what the horizon is by removing it.

Point the camera straight down and there is no horizon at all: the ground plane’s points at infinity image nowhere, because no ray from the eye parallel to the ground pierces a picture plane perpendicular to the vertical. The camera’s horizon() returns nothing rather than a line at some enormous coordinate, which is a refusal rather than a special case.

And the fraction rule loses its subject: a figure standing directly beneath the eye is drawn at zero length, and the others are drawn as radial marks whose length grows with distance from the centre. Height has become position, which is a different picture with different information in it — and no measurement of height is available from it at all.

What a wrong horizon costs #

The measurement’s sensitivity to the horizon is worth one section, because the horizon is the input most often placed by eye and the error it introduces is not symmetric.

The height measurement reads a cross-ratio in which the horizon crossing is one of the four points. Move the horizon and that point moves, so the recovered height moves — and the sensitivity grows as the object’s image shrinks, because the four points crowd together and the cross-ratio’s derivative with respect to any of them grows.

The site’s error figure puts numbers on that from the other end: a 1.83 m object at 3 m is measured to 0.28% per pixel, and the same object at 201 m to 18.2% per pixel. A pixel of horizon error costs the same kind of amount, and for the same reason — near the horizon the whole object occupies few pixels, and the geometry is asking those pixels for a great deal.

So the practical ordering is: measure things that are close and large in the frame, find the horizon from long parallels rather than from a guess, and on a tilted picture take the trouble to find the vertical vanishing point rather than assuming the fraction rule.

The reading habit #

Three questions to ask of a photograph before using the horizon for anything:

Are the verticals parallel? If they are, the plane is vertical, and both statements hold. If they converge, only the incidence does — and the convergence locates the vertical vanishing point, which is the fourth point the repair needs.

Where do the ground’s parallels meet? That is the horizon, and it is the only construction that does not assume the answer.

And is anything of known height standing on the ground? If so, the eye height follows from the crossing — exactly, on any picture, tilted or not. The object has to be standing on the ground, which is the assumption most often violated without being noticed: a figure on a step, a sign on a plinth, a person on a kerb. Each of those is measured against the wrong plane, and the error is the height of whatever they are standing on, unreduced by any amount of care with the horizon.