Figures on a street that slopes

Of all the rules a drawing manual gives, the one about figures is the most used and the least qualified. Draw a crowd, and put every head on the horizon. It is quick, it is close to right in a great many pictures, and it is the reason a street scene sketched in ten minutes looks inhabited rather than assembled.

It is also a statement about a plane, and the plane is level. The horizon is at eye level when the picture plane is vertical, and equal-height figures on level ground are cut by it at the same fraction of their drawn height — the eye height over the figure height — however far away they stand. When that fraction happens to be one, the horizon passes through every head, and the rule as it is usually taught drops the fraction and keeps the picture.

On a street that slopes the heads are still exactly collinear, and the line they lie on is not the horizon. It is the street plane’s own vanishing line, and on a street rising at one in twelve it stands 58.9 pixels above the horizon. A draughtsman who puts the furthest figure’s head on the horizon instead has taken 1.46 metres off a figure 1.62 metres tall.

The condition is about the eye, not about levelness #

The general statement is shorter than the rule it replaces and it mentions no horizon at all.

A figure standing at a point $Q$ of some plane has its head at $Q + h\hat{\mathbf{y}}$, where $h$ is its height and $\hat{\mathbf{y}}$ is the world’s up direction. The image of that head is the point where the ray from the eye $E$ through it meets the picture plane, and it lands on the plane’s vanishing line exactly when the direction $Q + h\hat{\mathbf{y}} - E$ is parallel to the plane. Write $E' = E - h\hat{\mathbf{y}}$, the point one figure-height vertically below the eye, and the condition becomes $Q - E'$ parallel to the plane, for every $Q$ in it.

Which happens exactly when $E'$ lies in the plane. So the rule is: the heads of equal-height figures standing on a plane are collinear when the point one figure-height vertically below the eye is a point of that plane, and the line they lie on is that plane’s vanishing line. Nothing in it says level, and nothing in it says horizon.

It is worth noticing what kind of object the head line turns out to be. Each head images at the vanishing point of the direction from $E'$ to that figure’s feet — a direction lying in the plane — so the heads are not merely on the plane’s vanishing line, they are a sample of it, one point per bearing. The order of the heads along the line is the order of the figures’ bearings as seen from the point one figure-height below the eye, and two figures standing on the same ray from $E'$ put their heads at the same place on the line however far apart they are. The head line is the plane’s set of directions, drawn, and a crowd is a scatter of samples of it.

Two things follow immediately, and both are surprising against the taught version. The rule is not a fact about the ground — it holds on any plane the eye is correctly placed above, including a ramp, a bank, a stepped terrace and a sloping street. And it is not a fact about the figures being in a row, which is why the five figures above are deliberately scattered across the street as well as along it. Heads collinear because the feet were in a row would be a statement about the row, and would prove nothing.

The heads are collinear, and the line is the street’s own #

The picture is a poor instrument for a difference of fifty-nine pixels between two nearly-parallel lines, so the marks are drawn again on their own, with the vertical scale stretched and the horizon taken as zero.

Three separate claims are settled in that one drawing and they are worth pulling apart, because only the first is what the rule promises.

The heads are collinear, to 2.8e-14 pixels, which is the arithmetic floor of a double and not a tolerance. Collinearity is an incidence and incidences survive projection exactly; a residual anywhere above the floor would mean the condition on the eye had not been met.

The feet are not, and they spread over 111 pixels. That is the part a reader can check on any photograph of a hill, and it is what makes the collinearity of the heads non-trivial: the five figures are at five different heights in the world, and their tops still line up.

And the line is measurable without the camera. Fitting a straight line to the five drawn head marks — five points on a page, nothing else — recovers the street’s vanishing line, and the angle between that line and the horizon returns the grade as 4.7635 degrees. So the construction runs backwards as well as forwards. A photograph of five people on a hill contains the gradient of the hill, provided their heights are equal and the photographer’s eye was at that height above the road.

The taught rule’s cost is metres, not pixels #

Fifty-nine pixels sounds recoverable. It is not, and the reason is that a pixel near the horizon buys a great deal of world.

The mechanism is worth stating because it is counter-intuitive in exactly one respect. The gap between the two lines does not grow with distance — it is $f\tan\theta$, a constant of the picture, the same 58.9 pixels at every depth. What grows is what those pixels are worth. A figure $d$ metres up the street is drawn smaller in proportion to $d$, so a fixed displacement of its head is a fixed fraction of the picture and a growing fraction of that figure. In metres of the figure’s own height, the cost is $d\tan\theta$ and it is linear in the depth.

That is the same reciprocal every measurement in this collection runs into. Height recovery from one photograph is well conditioned near the camera and badly conditioned far from it, and the essay that measured its error found the same linear growth with distance for the same reason. Here the error is not a sensitivity to noise but a fixed mistake in the construction, and it grows at the same rate.

Where the rule runs out of figure entirely #

The linearity has a consequence sharp enough to be worth its own statement, because it is the difference between an inaccurate rule and an impossible one.

The cost is $d\tan\theta$ metres and the figure is $h$ metres tall, so at $d = h/\tan\theta$ the cost is the whole figure. Beyond that depth a head placed on the horizon belongs to a figure of negative height — the construction has put the top of the person below the bottom. On a street rising at one in twelve that happens 19.4 metres away, which is a short block, and the shallower the street the further out it is: at one in a hundred it is 162 metres and at one in seven it is under twelve.

The repair costs a draughtsman one line and no arithmetic. Two figures of the intended height, set out correctly anywhere on the street, determine the head line; every further figure is then drawn by joining its feet to the correct end of that line rather than to the horizon, and the construction is the familiar one with a different line in it. The line can also be had from the street itself, since it is the vanishing line of the street plane and the kerbs meet on it — which is the same object the ramp’s own horizon is found from, arriving here through the crowd instead of through the edges.

Nothing in the drawing announces this. A figure of negative height drawn by putting a head below a pair of feet is simply a smaller figure, upside down in a way no one looks for, and the picture reads as a crowd receding up a hill. The failure is total and invisible, which is the pairing this collection keeps finding in taught constructions — the rule that draws another room produces a perfectly good picture of a room nobody specified, and this one produces a perfectly good picture of a street populated by people of impossible height.

The control — a level street #

A rule that fails on a slope must be checked on the level, or the measurement is a claim about the machinery rather than about the rule.

Nought pixels, exactly, rather than a small number. That matters more than the sloping measurement does, because it is what makes the rule’s survival intelligible. A manual’s rule is not wrong on the case it was written for; it is exactly right there, and a draughtsman drawing courtyards, quaysides and market squares will never meet an input on which the taught rule and the general rule differ at all.

This collection has recorded the pattern often enough to name it. A condition tested only where it cannot fail is not a test, and a null result is worth something only when the experiment could have come out otherwise. The level street is the case where the two rules coincide; the sloping street is the case that separates them, and it separates them by a metre and a half.

The pixel gap belongs to the lens; the angle belongs to the street #

One more distinction has to be made before the fifty-nine pixels can be quoted anywhere, because half of that number is not about the street at all.

So the 58.9 pixels is a statement about this picture and this lens, and a reader measuring a photograph taken through a wider lens would find a smaller gap for the same street. At twelve degrees of tilt the same plane leaves its line 233 pixels from the horizon through a 35-degree lens and 87 pixels through an 80-degree one, on prints of the same size — a factor of nearly three, from the camera alone.

What is invariant is the angle between the two drawn lines, read back with the focal length, and that comes out as the tilt whatever the lens. Which is the shape of every recovery on this site: the pixels are the camera’s contribution and the angles and ratios are the scene’s, and separating them is what turns a picture into a measurement rather than an illustration.

What the level-ground rule actually says #

The taught rule is worth putting back beside the general one, in the form the essay that measured it uses, because the version in most manuals has already lost a term. This is from the essay on the horizon at eye level.

Reading the two rules together shows what the manual dropped and in which order. The exact level-ground statement is a fraction: the horizon cuts a standing figure at the eye height over the figure height. Specialising to figures of eye height gives heads on the horizon. Generalising from the ground to any plane gives heads on that plane’s vanishing line. The commonly printed rule is the specialisation without the generalisation, so it is right on one plane for one height and is offered for all planes and all heights.

The essay on the horizon and the fraction already separated the two halves of the level-ground version and found that the incidence survives a tilted picture plane while the fraction does not. This essay separates them the other way: the incidence is the half that generalises off the ground, and it generalises exactly.

Vertical, not perpendicular #

One detail in the derivation is easy to get wrong and is worth flagging, because getting it wrong produces a residual large enough to look like a real effect.

A figure standing on a slope stands upright. Its head is $h$ above its feet in the world’s up direction, not $h$ along the street’s normal — a person on a hill does not lean into the hill. Substituting the normal for the vertical changes the head position by a factor of the cosine of the slope, which on one in twelve is four parts in a thousand. That is invisible in the prose, invisible in the picture, and about ten orders of magnitude larger than the collinearity residual, so it swamps the measurement completely while looking like nothing.

The distinction is the same one the essay on the nosings of a stair makes about a flight: what points up the pitch and what points up. A staircase contains no surface along its pitch, and a street contains no upright along its normal. Both errors are natural because the plane is the thing being thought about, and in both cases the object standing on the plane does not know it is there.

What this does not settle #

The reach of the result is narrower than the rule it replaces, and the boundary is worth drawing rather than implied.

It needs the eye placed correctly. The heads are collinear when the point one figure-height below the eye lies in the plane, and not otherwise. Figures shorter than that are not collinear at all — a set of 1.28 m figures on a street the eye stands 1.62 m above misses collinearity by 37.0 pixels against the arithmetic floor, which is a refusal rather than a degradation. The rule does not become approximate as the condition is relaxed; it stops holding.

That refusal is what makes the collinearity worth quoting at all, and it deserves a sentence of its own. A residual of 2.8e-14 pixels is only evidence if there is an input on which the same routine returns something else, and thirty-four centimetres of figure height is enough to move it by thirteen orders of magnitude. So the measurement is a discrimination rather than a tolerance: the machinery is capable of saying no, and it says no to the nearest plausible arrangement — a street of people slightly shorter than the photographer. Any drawing manual’s rule that has never been shown an input it must reject is a habit rather than a result, and this collection’s own machinery is held to the same standard.

It needs the figures to be equal. Nothing in the construction knows how tall anybody is; it knows that they are all the same, and the head line is what that sameness looks like in a picture. A crowd of mixed heights has heads scattered about the line rather than on it, and the scatter is not noise — each departure is that person’s height difference, projected. Read the other way, that is a measurement rather than an obstacle, and it is the one a height out of one photograph makes on a single figure using a reference. What the head line adds is that the reference can be the crowd.

It needs one plane. Figures on a street and figures on the pavement beside it, at a different grade, lie on two different vanishing lines, and nothing joins them. A picture of a hill with a level terrace cut into it has two head lines and a draughtsman has to know which figure is on which.

It gives no height. The line says the figures are equal and says nothing about what they are equal to; the metric statement needs a reference length imported from outside the picture, exactly as carrying a height across a room does. What the drawn line supplies is the shape of the arrangement, which is all a projection ever supplies.

And it says nothing about whether a viewer notices. A crowd drawn with its heads a metre and a half short at the far end is a crowd of children at the far end, and reading it as such is a fact about seeing rather than about geometry.

One plane at a time, again #

The finding belongs beside two others rather than in a chapter on figure drawing.

The ramp has its own horizon established that a tilted plane’s vanishing line is a real line in the picture with the gradient readable off it. The measuring point moved onto a ramp showed that the metric constructions have to be anchored to that line rather than to the ground’s. This essay adds the third: the rule about figures is anchored to it too, and the head line of a crowd is that plane’s vanishing line made visible by people.

Which suggests reading the whole taught vocabulary the same way. Wherever a construction says the horizon, it means the vanishing line of the plane this construction is about, and the substitution is invisible because almost every demonstration is drawn on the ground. A drawing has three horizons says as much about the three coordinate planes at once; a street with a camber and a rising grade has as many as it has surfaces. The eye-level horizon is special only in that the ground is where things stand — and on a hill, they stand somewhere else.