The eye that reaches the most

What an eye can paint establishes that the part of an object a design reaches depends on where the eye is. The obvious next question is which eye is best, and the obvious answer — as high as possible, since a higher eye sees over things — turns out to be right on three objects out of four and wrong on the fourth for an instructive reason.

More usefully, the question is malformed until a price is named.

Two quantities, moving opposite ways #

A design cast from an eye onto an object has two obvious measures of success, and they are not the same measure.

How much of the object gets picture. The share of the surface that receives any of the design, which is what a designer with an object to decorate wants to maximise.

How much of the design lands. The share of the intended picture that reaches the object at all, rather than sailing over it into the room, which is what a designer with a picture to show wants to maximise.

A higher eye improves the first. It sees over whatever was in the way, and the occlusion clears entirely at a computable height.

A higher eye can worsen the second. Rays from a high eye through a design are shallower, so they travel further before they meet anything — and past a certain height they clear the object’s far end and land nowhere.

Whether the two conflict depends on whether the object has a far end within reach, and that is the whole of the difference between the four cases below.

Three objects where higher is better #

A cluster of blocks. From 1.2 metres to 5, the surface reached goes from 71 per cent to 80 and the share of the design landing goes from 91 to 99. Both rise. The design’s stray rays land on the floor beyond the blocks rather than escaping, so there is no cost to raising the eye and the answer is the top of the range.

A seating rake. Surface from 47 per cent to 90, design landing from 100 to 98. A tiny cost against a very large gain, and again the answer is as high as the room allows.

An ascending flight. Surface from 64 per cent to 93, design landing essentially constant at 100.

On all three the honest advice is stand higher, and the reason is the same each time: the object is backed by more object, so a ray that misses one face hits another.

The corridor, where they cross #

The corridor is the object with a far end, and the far end changes the answer.

Its surface reached is 98.9 per cent at 1.2 metres and reaches 100 by 2.2, so there is almost nothing to gain — the corridor’s walls are all reached from any eye that can see into it. Its design landing is 100 per cent up to 2.7 metres and then falls: 98 at 3.3, 95 at 4, 92 at 5.

The mechanism is the doorway and the wall beyond it. Rays through the opening travel nine metres before they hit anything, so a small increase in the eye’s height sends them over the far wall entirely; and rays that used to hit the end wall above the doorway go over it.

Multiply the two — the share of the object reached times the share of the design that lands, which is a rough measure of picture actually delivered — and the product has a genuine interior maximum at about 2.2 to 2.7 metres, falling away on both sides.

And how far back, which behaves completely differently #

Height is one of the eye’s coordinates and distance is another, and sweeping the second gives a different shape with a different explanation.

Hold the eye at 1.7 metres and move it from 1.2 metres in front of the intended picture to 5.5. On a cluster of blocks the surface reached goes 74, 76, 73, 72, 73, 69 per cent — a shallow maximum at about 1.8 metres and a slow decline after — and the number of faces reached goes from nine, up to ten, and down to five.

The mechanism is not occlusion. It is the design’s own cone: standing further back, the same design subtends a narrower angle, so its rays are more nearly parallel and sweep across less of the object. A design an arm’s length away spreads over everything nearby; the same design from five metres is a narrow pencil that lands on a few faces and stretches them less.

The trade is therefore between coverage and stretch, and it runs the other way from the height trade: the worst local stretch falls from 22 to 17 as the eye retreats, so a design cast from further back covers less and covers it better.

On a corridor nothing moves at all — every distance reaches the same six faces and lands all of the design — because a corridor’s walls run away from the eye and there is nothing for the narrowing cone to miss.

Where the interior optimum really is #

Two of the sweeps have no interior best and one does, and the distinction has a general form worth stating — after which the distance sweep turns out to have an interior best that the essay above reads as a monotone decline.

Coverage saturates and landing declines. The share of the object reached is bounded above by one and rises toward it; the share of the design that lands is bounded above by one and falls once the object’s far edge is inside the design’s cone. So the product has an interior maximum exactly when the object has a far edge within reach, and not otherwise. A cluster of blocks standing on a floor that runs on has no far edge, so both curves rise together and the answer is the top of the range; a corridor ends, so it has one. That is the whole of the difference between the four cases, and it can be decided from a plan without sweeping anything.

The distance sweep has one too, and it is not at either end. For a mark at depth $d$ beyond the picture plane, the stretch is $(d + e_z)^{2}/h\,e_z$, and differentiating with respect to the eye’s distance gives $(d+e_z)(e_z - d)/e_z^{2}$ — zero at

$$e_z \;=\; d.$$

Stand as far in front of the design as the marks lie beyond it. For the cluster, whose far marks are around three metres out, that is three metres back, which is inside the sweep’s own range and is why the stretch falls from 22 to 17 across it rather than falling indefinitely. Past the optimum it rises again, slowly, because the picture is being cast almost parallel and the marks are running out along the floor.

So the honest summary of the two coordinates is not that height has an optimum and distance does not. Both do; the height’s is a property of the object’s extent, and the distance’s is a property of the design’s own reach and sits at a place a designer can compute in one line.

The claim that had to be withdrawn #

The first version of this measurement asserted that the best eye is an interior height on every object, and the assertion passed. It was wrong, and the way it was wrong is worth publishing because it is a trap any sweep can fall into.

The surface share does not rise smoothly and then fall. On the cluster of blocks it rises and then plateaus: 77 per cent at 2.5 metres, 77 at 3, 77 at 3.8. A search for the maximum over a sampled sweep returns the first of several equal values, which is an interior sample — and a test asking “is the best an interior height” returns yes.

It means nothing. Ties are not peaks, and the first of a plateau is an artefact of the sampling order rather than a property of the object.

The measurement now counts the ties and reports whether the maximum is strict, and the corrected claim is the one above: three objects with maxima at the end of the range, one with a real interior optimum, and one methodological reason to distrust an argmax reported without its ties.

That is the same species of error as a residual maximised over two families and a conformality measured where it cannot fail: a summary statistic that is technically correct and answers a question nobody asked.

And a third quantity that reverses the ordering again #

There is a third thing a designer might maximise, and it puts the answers in yet another order.

How much of the design is paintable — the share that lands and arrives at a local stretch a painter would accept. The stretch decides the band is the rung that measures the cap; taking four as the cap, the sweep gives:

On the flight, 87 per cent at 1.2 metres rising to 100 at 4, with a dip at 1.8 on the way.

On the corridor, 73 per cent rising to a maximum of 92 at about 3 metres and then falling — an interior optimum again, but at a different height from the one the product gave.

On the rake, 83 per cent at 1.2, up to 87 at 1.5, down to 81 at 2.2 and back to 89 at 5. Non-monotone with an interior minimum, because at middling heights the design lands on treads at a grazing angle and at low or high ones it does not.

So the same object has its best eye at 5 metres on one measure, at 3 on another, and at 1.5 on a third, and none of those is a mistake. They are answers to three different questions.

The rake’s dip, which is a real object #

One row in the sweep is worth pausing on because it looks like noise and is not.

The rake’s surface share goes 47, 47, 61, 52, 66, 85, 81, 81, 90 as the eye rises. The dip at 1.8 metres is a face lost and regained: at that height one tread high up the rake is grazed by the design’s last rays and receives a handful of marks, then loses them, then gains a much larger share when the eye is high enough to see it properly.

A face that receives four marks out of six hundred is “reached” by the measurement’s own definition, and whether it counts is a threshold nobody set. That makes the surface share slightly jumpy near any eye height where a face is about to be released, and it is worth knowing before reading too much into a single sample.

The fix is not to smooth it. It is to report the share of the design a face receives rather than a yes-or-no, which is what the per-face shares do, and to treat the face count as the coarse summary it is.

What this means for a real design #

The four objects give one piece of advice each and one general one.

For a stack of things — blocks, steps, a rake — stand as high as the room allows. Everything improves and nothing is lost.

For an object with a hole in it, find the height where the rays through the hole still land, and stop there. That height is computable in advance from the depth beyond the opening and the size of the design.

For anything, say which quantity is being bought before asking for the best eye. Surface covered, picture delivered, and picture paintable are three answers, and on the corridor they differ by two metres of eye height.

And check the ties. A maximum reported from a sampled sweep without its plateau is a statement about the sampling.

Why this is not the same question as the tolerance #

Two rungs of the anamorph field now measure something about where the eye is, and they are easy to run together and should not be.

The room the eye may stand in asks: given a design already made for one eye, how far may a reader move before the picture stops working? That is a question about a tolerance, its answer is a spindle of thirty-one cubic centimetres, and the design is fixed.

This rung asks: where should the design’s eye be placed in the first place, so that the design covers as much of the object as possible? The design is not fixed — it is redrawn for each candidate eye — and the answer is a metre or two of height rather than a few centimetres of tolerance.

The two are different by three orders of magnitude in size, which is the clearest sign they are different questions. A designer chooses the eye once, from a range measured in metres, and then everybody who comes to look has to stand within a few centimetres of it.

That asymmetry is the anamorph’s whole practical difficulty, and it is worth saying plainly: the choice is generous and the consequence is not.

One more reason the question needs a price #

There is a fourth thing a designer might want and this collection has measured it already, which is why it is worth naming here rather than sweeping it.

The eye that is easiest to occupy. An anamorph’s design eye becomes a place people have to stand, and some places are easier to stand in than others — a viewing platform at four metres is a building project where an eye at 1.65 is a floor marking. An anamorph at true size is the collection’s one unconditional viewing-distance claim, and it exists precisely because a design’s eye has to be somewhere a body can be.

So the full list of quantities a designer chooses the eye on is four: surface covered, picture delivered, picture paintable, and eye reachable. The first three are geometry and this rung measures them; the fourth is architecture and decides more real designs than the other three together.

That is not a complaint about the measurement. It is the reason the measurement is worth having in the form of curves rather than a single best height: a designer told “2.4 metres” has been given an answer they may not be able to use, and a designer shown three curves can find the best eye among the heights a person can actually occupy.

Why the sweep is over height rather than over everything #

The eye has three coordinates and this rung sweeps two of them, which leaves the third worth a sentence.

Sideways is the one not swept, and it is not swept because on every object here it is a symmetry. The blocks, the rake, the corridor and the flight are all symmetric about their own centreline, so an eye off to one side gives a mirror image of an eye off to the other and the maximum is at the middle by construction.

That would stop being true the moment an object is asymmetric — a corridor with a doorway to one side, a cluster of blocks arranged unevenly — and the measurement would then have a genuine third dimension in it. It is not run here because none of these objects has that property, and running it would produce a flat curve and an assertion that the flat curve is a finding.

Which is worth naming as a habit rather than as an excuse: a sweep over a parameter the arrangement is symmetric in measures the symmetry. The collection has at least one figure whose subject turned out to be exactly that — a taught two-point cube reporting a side ratio of exactly 1.000 at every setting, because moving both far edges together keeps the drawing symmetric and symmetry forces a square.

The short version #

A higher eye buys the faces occlusion was hiding and spends design off the far end of the object. On three of four objects nothing is spent, because the object is backed by more object, and the best eye is the highest available.

On a corridor with a doorway the two cross, and the best height depends on which is being bought: 2.2 to 2.7 metres for picture delivered, 4 metres for picture paintable, and no height at all for surface covered, which is flat from two metres up.

The first version of this measurement said every object had an interior optimum. It was reading the first of a plateau as a peak.