Two distances to infinity

The eye taken to infinity makes the argument that a parallel projection is what a perspective one becomes when the centre recedes, and it makes it without a number. “Far enough away” is the phrase, and it is the phrase this essay is about.

There is a number. There are two of them, and they differ by an order of magnitude, which is the finding.

The measurement #

Take an object and a direction. Cast a design along the direction — the parallel case — and then cast the same design from an eye standing at each of a sweep of distances along that direction, and compare the two mark by mark.

Two things can be compared and they behave completely differently.

Which faces are painted. The set of faces the design lands on, against the set the direction lands on.

Where the marks land. For each design point that lands on the same face both ways, how far apart the two landings are in metres.

The faces settle almost at once #

Three object radii. At that distance the eye is painting exactly the set of faces the direction paints, and it never differs again.

The reason is that the set of faces is a question about the silhouette. A face is round the back or it is not, and which it is changes only when the eye crosses the plane of that face — an event that happens at a distance set by the object’s own shape, not by any tolerance. Once the eye is outside every face’s plane in the same sense the direction is, the sets agree exactly, and moving further changes nothing.

So this half of the limit is not asymptotic at all. It arrives, at a distance, and stays.

And the marks take an order of magnitude longer #

Thirty-four object radii, against a design sample of about thirty centimetres on this object.

The displacement falls like one over the distance, which is the expected law and is checked rather than assumed: over the tail of the sweep, from thirty-four radii to four hundred, the shift falls by a factor of 12.4 while the distance grows by 11.8 — a ratio of 1.06 against the 1.00 an exact reciprocal gives.

At three radii — where the faces have already settled — the marks are still 0.83 metres out of place, which is nearly three design samples. At thirty-four they are inside one.

Why the two differ so much #

The two halves are asking different kinds of question, and the difference is not a matter of one being harder.

The faces are a discrete question. The answer is a set, it changes by whole faces, and it changes at specific distances. Between those distances the answer is exactly right rather than approximately.

The marks are a continuous one. The answer is a position, it moves smoothly, and it approaches its limit without reaching it. There is no distance at which a mark is exactly where the direction would have put it, only a distance at which the difference is below whatever the design cares about.

That is the structure of nearly every limit in this collection and it is usually invisible because only one half is measured. Here both are, and the honest statement of “a parallel projection is the limit of a perspective one” is that one of its consequences arrives and the other converges.

What “far enough away” is for #

The number matters because the question is asked in practice, and it is asked in two different situations that want the two different answers. The eye that reaches the most asks a related question of a near eye, where the answer has an interior optimum rather than a limit.

A drawing that needs the right faces. A cutaway or an exploded assembly cares which surfaces are visible and does not care about a millimetre of position. Three object radii is enough, which is a very ordinary photographic distance — an object a metre across, photographed from three metres.

A drawing that needs the right places. A measured drawing, a template, a set-out for a marking is a statement about where things are, and thirty-four radii is a photograph of a metre-wide object taken from thirty-four metres. That is a long lens across a room, and it is why a technical photograph intended to be measured is taken that way rather than for any reason about perspective looking wrong.

The design sample is the tolerance #

The mark distance depends on what counts as close, and here the criterion is one design sample — the spacing of the grid the design is written on.

That is the right criterion and it is worth saying why rather than picking a millimetre. A design is a picture made of marks, and two arrangements that put every mark within a sample of each other produce the same picture at that resolution: there is no observation the design supports that can tell them apart. A tolerance in metres would be a statement about the object; a tolerance in design samples is a statement about the picture, which is what is being compared.

It also means the answer scales the way it should. A finer design has a smaller sample and needs a longer distance; a coarse one needs less. The thirty-four radii quoted here is for a twenty-one by twenty-one design, and a design at twice that resolution needs about twice the distance — exactly twice, since the mark displacement falls as $1/D$ and the tolerance falls as $1/n$, so $D \propto n$ with no other term in it.

The order of magnitude between the two distances has the same kind of explanation, and it is a general one. Which face a ray strikes is a discrete quantity and where on it the mark lands is a continuous one. A discrete answer stops changing once the rays have swung by less than the angular size of the smallest face — a large quantity — while a continuous one keeps changing until the displacement falls below a tolerance somebody chose. So the faces settle at a feature and the marks at a tolerance, and the gap between the two distances is the ratio between a feature and a tolerance rather than anything about projection.

The arithmetic behind the reciprocal #

The law is checked rather than assumed, and it is worth knowing where it comes from because the exponent is the thing most easily got wrong.

A perspective ray from an eye at distance D through a design point differs in direction from the parallel ray by an angle of order (object size)/D. Over the object’s own extent that angular difference produces a displacement of order (object size)²/D. So the shift falls like one over the distance, with a coefficient that is the object’s size squared.

Not one over the square. That would be the law for a quantity measured at the eye rather than on the object, and it is the answer a reader who has met the inverse-square law in other contexts will reach for. Here the displacement is on the object, the object does not move, and one power of the distance is all there is.

The fit confirms it: over the tail of the sweep the ratio comes out at 1.06 against the 1.00 an exact reciprocal gives. It is fitted on the tail rather than across the whole sweep because near the object the comparison is over a changing set of marks, for the reason given below.

What a long lens actually buys #

The practical form of the question, since a photographer’s version of “far enough away” is a focal length rather than a distance.

Standing thirty-four object radii back and framing the object fills the frame at a focal length of roughly thirty-four times the sensor’s own half-width — which for a full-frame sensor is about six hundred millimetres. That is a long lens, and it is the lens a technical photograph of a small object intended to be measured is actually taken with.

For the faces alone, three radii is about a fifty-millimetre lens on the same sensor, which is a normal one. So the two answers correspond to two entirely different photographs: an ordinary picture of an object is already parallel enough for a cutaway, and is nowhere near parallel enough for a template.

That is a useful thing to be able to say, and it is a number rather than a rule of thumb. The screen sets the distance makes the same kind of translation at the other end of the chain — from a focal length to a place in a room — and the two together turn a claim about limits into a claim about equipment.

What settles even faster #

There is a third quantity, and it settles before either — worth naming because it is the one a reader is most likely to have in mind.

The shape of the drawn outline. A perspective projection’s outline converges on a parallel one’s rather quickly, because an outline is a large-scale feature and the leading correction to it is uniform. That is why a photograph of a building from across the street already looks nearly axonometric and why the eye is a poor judge of this limit.

So there are three arrivals, at three different distances, and a reader who has satisfied one may assume they have satisfied the others. The outline is right long before the faces are, and the faces long before the marks. Any statement of the form “far enough away to be parallel” therefore has to say which of the three it is about, and almost none do.

The objects differ, and how they differ is informative #

The sweep runs on several objects and the two distances move.

A corner settles its faces immediately, at the first distance sampled, because a convex object has no occlusion to change. A cluster of blocks takes longer, because which block hides which changes as the eye moves. A corridor with a doorway takes longest, because a doorway is a hole and whether a ray passes through it is exquisitely sensitive to the ray’s direction.

The mark distance moves much less between objects, because it is set by the arithmetic of the divisor rather than by the object’s topology: it is essentially the object’s own size over the tolerance, and every object here is about the same size.

That is a clean split and it is worth having. The face distance is a property of the object’s shape; the mark distance is a property of the object’s size and the design’s resolution. Neither is a property of the projection, which is what makes them both measurable rather than definitional.

Why the mark comparison drops some samples #

A measurement detail that changes the number, so it is stated rather than buried.

A mark is compared only where the two casts land on the same face. A design point that has slid onto its neighbour across a crease has moved by the width of the crease rather than by the amount the limit is about, and including it would report the object’s corners rather than the convergence.

That refusal is why the near end of the sweep is noisy: at one and a half radii many samples are dropped, so the comparison is over a changing set of marks as well as a moving one, and the reciprocal law is fitted on the tail rather than across the whole sweep for exactly that reason.

Where the limit is met in the rest of the collection #

The same convergence appears in three other places here, and putting them side by side says something the individual cases do not.

A stitched panorama. A rig is right on one surface measures how far away a scene has to be before a camera rotated about the wrong point stitches without ghosting, and the answer is a distance rather than a limit: the parallax falls like one over the scene’s distance and the tolerance is a pixel.

A light far enough away. A light far enough away asks when a lamp becomes the sun, and finds that the crossover is a number rather than a philosophical boundary — the shadow lines’ convergence falls off and the distance at which it stops being detectable is measurable.

And a screen far enough away. The screen sets the distance turns a focal length into a place in a room, and the whole screen field is about how rarely anybody is standing at it.

All four are the same shape: a projective idealisation, a real arrangement that approaches it, and a tolerance that turns “approaches” into a number. The habit worth carrying is that the tolerance is always the interesting part, because it is the only thing in the statement that is about the observer rather than about the geometry — and it is the part that is almost always left out.

What this says about the field’s founding claim #

The parallel field opens by insisting that parallel projection is not primitive perspective — not a technique used by people who had not worked out the real one, but a different answer to a different question. Parallel projection is not primitive perspective makes that case on what each system preserves.

This essay is the other half of it, and it cuts the same way. A parallel projection is a limit of perspective ones in a precise sense, and the limit is not reached at any finite distance for the quantity a measured drawing cares about. So a parallel drawing is not a perspective drawing taken far enough away; it is the thing at the end of a sequence that has no last term, and it has exactly the property — invariance under a slide along the direction — that no member of the sequence has.

That is why nothing moves along the direction is an equality in a parallel system and an approximation at every finite distance, however large.

A reading a reader can take #

The measurement is synthetic and the question it answers is one somebody may have about a real photograph, so it is worth saying how the two connect.

A reader with a photograph of an object and a suspicion that it is “near enough parallel” has one thing available: the object’s own parallel edges. In a parallel projection they are parallel on the page; in a perspective one they converge to a vanishing point, and how far off the page that point is is exactly the distance in question.

So the reading is: extend two edges that are parallel in the world, find where they meet, and divide that distance by the object’s own drawn size. A meeting point three object-widths off the page is an eye three radii back — the faces have settled and the marks have not. A meeting point thirty widths off is thirty radii, and the marks have settled too.

That is a construction with a straightedge and no arithmetic, and it converts the two numbers here into something a reader can check on a photograph they hold. Both vanishing points on the paper is where this collection establishes that a vanishing point off the page is perfectly usable, which is the fact the reading rests on.

The short version #

Which faces a design paints settles at three object radii and never changes again, because a face is either round the back or it is not. Where the marks land falls like one over the distance and takes thirty-four radii to come inside one design sample.

“Far enough away to count as parallel” therefore has two answers an order of magnitude apart, and a third — the outline’s shape — that arrives before either. A drawing that needs the right faces is finished long before a drawing that needs the right places, and one face, one scale is the property that never arrives at any finite distance at all.