Where the anamorph still works

An anamorph is described as a picture that is correct from one point. Taken literally that is a statement about a set of measure zero, and it cannot be what anybody means: a viewer occupies a volume, has two eyes sixty-three millimetres apart, and breathes. Something must be true about a neighbourhood of the design viewpoint, and the description does not say what.

The rung below this one shows that a floor anamorph is a planar homology built out of three numbers, all three of which are the eye’s coordinates. That immediately gives a way to ask the question properly. Two eyes give two maps; a viewer standing in the wrong place — which is the ordinary condition rather than the exceptional one — sees the intended design composed with one map and un-composed with the other, and the composition is an object with a name.

The error map has a form #

Write $H_E$ for the homology carrying the rabatted design to the floor marks from the design eye $E$, and $H_{E'}$ for the one belonging to a different eye $E'$. A viewer at $E'$ reads the marks by running that second map backwards, so the picture they get is the intended design composed with

$$H_{E'}^{-1} H_E$$

Both factors have the ground line as their axis, and the maps of a plane that fix one line pointwise form a group. So the composition fixes the ground line pointwise too — always, for every wrong eye, without any assumption about how far it moved or in what direction.

That is the whole of the result, and everything else in this essay is a consequence of it.

The error is exactly zero along the line the picture stands on. Not small, not first-order: zero. A pavement anamorph’s bottom edge is right from anywhere in the room, and what falls apart is everything above it.

Three moves, three answers #

Which central collineation it is depends only on how the eye moved, and the three axis-aligned moves give three different answers with closed forms in the two eyes and nothing else.

Sideways, parallel to the picture, is an elation. No point off the ground line is fixed and there is no characteristic ratio, because the centre has fallen onto the axis. What the reader sees is the design sheared: each row of it slid sideways, by an amount proportional to its height.

Up or down is a homology whose centre is at infinity. A centre at infinity means the lines along which points slide are all parallel — vertical, here — so the map is an affine stretch of the picture’s height, with ratio the ratio of the two eye heights. Raise the eye by a sixth and everything in the reconstructed picture is a sixth taller.

In or out is a homology with a finite centre, and the centre is the eye’s own height and lateral position on the picture plane. The picture is scaled about the point directly in front of the reader, by the ratio of the two distances. That is what walking toward any picture does, and it is the sense in which the line of sight is the forgiving direction.

The two costs are the same number #

Drawing the sideways curve and the vertical curve on the same axes gives two lines that lie on each other, and for as long as it took to write the algebra out that read as a bug.

It is not. Displace the eye sideways by $\delta$, and the reconstruction of a design point at height $y$ slides sideways by

$$\delta \cdot \frac{y}{e_y}$$

its own height above the ground line, over the eye’s height. Displace the eye upward by the same $\delta$, and the same design point rises by

$$\delta \cdot \frac{y}{e_y}$$

which is the same expression. The two costs are equal to the last bit — the departure from the closed form over four heights is under $10^{-16}$ metres in both cases — and they differ in direction and in kind rather than in size. One is a shear and one is a stretch; the reader’s eye is far better at noticing one of them than the other, and the geometry does not care which.

For the figures here, with the eye 1.62 m up and a design 1.38 m tall, a step of a quarter of a metre in either direction puts the top of the design 212.5 mm from where it should be. That number is worth carrying because it is the tolerance stated properly: a step of $\delta$ moves the top of an anamorph by $\delta$ times the ratio of the design’s height to the eye’s. A design that reaches the viewer’s own eye level is displaced by the full step; a design half that tall is displaced by half of it.

Where the two expressions come from #

Both fall out in three lines, and writing them out is worth the space because the equality is otherwise a coincidence a reader has to take on trust.

A design point $(x, y)$ casts its mark at parameter $t = e_y / (e_y - y)$, so the mark sits at $\big(e_x + t(x - e_x),\ e_z(1 - t)\big)$. Reading that mark back from an eye $E'$ uses the mirror-image parameter $s = e_z' / (e_z' - Z)$, where $Z$ is the mark’s depth.

For a sideways move the two eyes share a height and a distance, so $Z = e_z(1 - t)$ gives $s = 1/t$ exactly. The reconstructed height is then $e_y(1 - 1/t) = y$: unchanged, which is why a sideways step does not move anything vertically. The reconstructed lateral position works out to $x + \delta(1 - 1/t)$, and $1 - 1/t$ is precisely $y / e_y$.

For a vertical move the distance is again shared, so $s = 1/t$ once more and the lateral position is unchanged. The height becomes $(e_y + \delta)(1 - 1/t) = y\,(e_y + \delta)/e_y$, which is $y + \delta y / e_y$.

The same factor $1 - 1/t = y/e_y$ appears in both because both moves leave the depth of every mark alone, and the depth is the only thing $t$ depends on. A move in or out changes the depths, $s$ is no longer $1/t$, and the clean factor disappears — which is why the third curve has a different shape rather than a different slope.

The third curve, and why it is the cheap direction #

The in-or-out case has a closed form too, and it is worth writing because it turns “the forgiving direction” into a factor.

Reading a mark from an eye at distance $e_z' = r\,e_z$ reconstructs a design point at

$$y' \;=\; \frac{e_y\,y}{r\,(e_y - y) + y},$$

a Möbius map in $y$ whose two fixed points are $y = 0$ and $y = e_y$ — the ground line, as the group argument requires, and the eye’s own height, which is the homology’s finite centre. For a small step $\delta$, with $r = 1 + \delta/e_z$, the departure is

$$y' - y \;\approx\; -\frac{\delta}{e_z}\cdot\frac{y\,(e_y - y)}{e_y},$$

a parabola — zero at both ends and largest in the middle, at $y = e_y/2$, where it reaches $\delta\,e_y/4e_z$. That is the shape the third figure shows, derived rather than described, and it explains the one feature the other two curves do not have: the departure comes back down, because the map is a scaling about a point inside the picture rather than about a line below it.

That is only the vertical half of what an in-or-out step does. The same step also spreads the picture sideways: the reconstruction’s lateral coordinate is multiplied by $1 + \delta y/e_y e_z$, so a design point $x$ from the centre moves by $|x|\,\delta\,y/e_y e_z$. The two components combine to

$$\frac{\delta}{e_z}\cdot\frac{y}{e_y}\sqrt{(e_y - y)^{2} + x^{2}},$$

whose largest value is at a corner of the design rather than in the middle of it, and which for a design of any width is dominated by the lateral term.

Put the worst cases side by side. A step $\delta$ across the sightline costs $\delta\,y_{\max}/e_y$ at the top of the design. The same step along it costs the expression above at the design’s far corner — about a third of that on ordinary proportions, and never the fifth that the vertical term alone would suggest.

Three things follow, and the first is the one a designer can act on.

The acceptable region is a spindle, about three times longer along the line of sight than across it, which is the shape the room the eye may stand in measures by search and finds at 2.56 — and the factor is now a formula rather than a number, so it can be computed for any arrangement before the search is run.

The advantage grows with the eye’s distance and shrinks with the design’s width. A reader standing far back from a tall design has an enormously elongated region; one standing close to a short one has a nearly round one. A reader standing far back has an elongated region; a wide design pulls it back toward round, because the lateral spreading is proportional to how far from the centre the design reaches. A tall narrow anamorph is the forgiving one along the sightline, and a wide low one is not — which is the opposite of the ordering the design’s own trade sets up on cost.

And it is the ratio, not the direction, that a viewer experiences. Both costs are linear in $\delta$, so no step is safe and none is catastrophic; what the geometry says is only that walking toward the picture is worth about three steps of standing to one side of it, which is why standing in the wrong place is tolerable for an ordinary picture and why an anamorph’s reader is told where to stand rather than which way to face.

Along the sight line is cheaper, and by a factor #

The three moves are not equally expensive, and the difference is a factor rather than a rounding.

Over the same quarter-metre step, the worst departure across the design is 212.5 mm for a sideways or a vertical move and 43.3 mm for a move in or out — a ratio of 4.91 on this eye and this design. So the tolerance of an anamorph is a shape: a long thin volume with its length along the sight line.

The reason is visible in the classification. Moving in or out scales the picture about a point inside it, so the departure vanishes at that point and grows away from it in both directions, reaching a maximum in the middle of the design’s height and falling again. Moving sideways or upward slides or stretches about the ground line, which is at the bottom of the design, so the departure grows monotonically all the way to the top.

Two eyes are two wrong eyes #

A viewer has two of them, sixty-three millimetres apart, and the analysis above already covers the case: each eye is a sideways displacement of 31.5 mm from the point between them, in opposite directions.

So the two eyes see the design sheared by $\pm 31.5\,\text{mm} \cdot y / e_y$ — equal and opposite. At the eye height and design height used here that is 26.8 mm of shear at the top of the design for each eye, in opposite directions, which is a disagreement of 53.7 mm between what the two eyes report about the same painted mark.

That disagreement is a fact about the painting rather than about the viewer, and it is the reason a flat anamorph reads as a flat surface with an odd pattern on it rather than as the thing it depicts. The marks are on a plane, both eyes agree about where the plane is, and the depicted scene is the only thing they disagree about. A painted anamorph cannot be built to satisfy two eyes at once, because it has one surface and they want two different ones. The set built in three dimensions is the construction that tries, and the rung there measures what it achieves and what it gives away.

Where the map stops existing #

There is a limit past which none of this applies, and it is not a gradual one.

Reading a mark back to the picture plane needs the mark to lie beyond that plane from the reader’s position. A reader who walks past the ground line — who steps over the painted edge and turns around — has marks on both sides, and the ones behind produce no picture point at all rather than a badly placed one. The machinery refuses there, and the refusal is checked in the site’s own gate rather than assumed: a mark at exactly the reader’s own depth has no reading, and one a hair beyond it does.

The other limit is the reader’s height. The design’s own construction refuses any point at or above the design eye’s height, so a reader who crouches below the ground line’s own plane is not reading a distorted anamorph; there is nothing there to read. Both limits are edges of the map’s domain, and both are places where a routine that returned a number anyway would produce a picture of somewhere nobody is standing.

Against an ordinary picture #

None of this makes an anamorph a special kind of object, and the comparison is the reason to say so.

An ordinary photograph is also correct from one point — the point at the picture’s own focal length, scaled to the width it is printed at, which is what this site computes for every figure it draws. Reading it from somewhere else also composes it with a projectivity. The difference is the size of the numbers, not their kind: for a photograph held at arm’s length the ratio of the design’s extent to the viewing distance is small, and for an anamorph on a floor it is not.

What a painter does about it #

Two consequences follow for anybody actually laying one of these out, and both are geometric rather than practical advice.

Put the axis where the reader looks. The picture is exactly right along the ground line and wrong in proportion to height above it, so an anamorph whose design stands against a wall, a step or the far edge of a court gives the reader a correct edge to anchor on. One painted with its ground line in the middle of an open floor has its most reliable part where nobody is looking.

Keep the design short relative to the eye. The departure scales as design height over eye height, so halving the design’s height halves the error for every possible misplacement at once. A pavement piece whose figures rise to the height of a standing viewer is at the worst possible ratio: a step of $\delta$ moves the top by $\delta$.

Neither of those is advice about taste, and that is the point of stating them as geometry. The first says where to put a line; the second says what ratio to build at. Both are consequences of the axis being fixed pointwise and of the departure being linear in height, and both would be invisible to anyone describing an anamorph as a picture that resolves from one spot.

There is a third consequence that reads as a paradox and is not. Moving the design viewpoint further back makes the anamorph more forgiving of a sideways step and less forgiving of a step in or out. The sideways cost is $\delta y / e_y$ and does not contain the distance at all; the in-and-out cost scales the picture by the ratio of two distances, and that ratio is nearer one for a distant eye. So a design laid out for a far viewpoint tolerates a step forward better and a step sideways exactly as badly, and the shape of the tolerance volume gets longer rather than fatter.

The short version #

A wrong viewpoint does not give a wrong picture; it gives the intended picture composed with a central collineation whose axis is the line the picture stands on. So the picture is exactly right along that line, and wrong above it in proportion to height.

A sideways step of $\delta$ shears the picture by $\delta y / e_y$. An upward step of $\delta$ stretches it by exactly the same amount. A step of $\delta$ along the sight line costs about a fifth as much, because it scales the picture about a point inside it rather than about its bottom edge.

That is what “correct from one point” means once it has a number attached: correct on a line, wrong in proportion to height, and five times more forgiving forward and back than side to side.