Anamorphosis is only a viewpoint

Holbein’s The Ambassadors has a grey smear across its lower third that means nothing from in front of the painting. Move to the right and low, close to the wall, and it resolves into a skull.

The usual account treats this as a trick — a piece of virtuosity bolted onto an otherwise ordinary picture. It is not. The skull is drawn in perspective, correctly, with exactly the same construction as the rest of the painting. The only difference is that its centre of projection is not where the picture’s is.

The construction #

An anamorphic image is made by choosing a viewpoint, choosing a surface, and projecting.

Take the picture that is meant to be seen. Position it in space where it would look right — a small upright rectangle in front of the viewer. Choose the surface it is actually going to be painted on, which is at some steep angle to that rectangle. Project from the chosen viewpoint through the rectangle onto the surface.

The result is a stretched, sheared, unrecognisable version of the original, and from the chosen viewpoint it is indistinguishable from the original, because every ray from the eye passes through both.

For a flat surface the whole operation is a single homography: an eight-parameter map of the plane to itself, determined by where four points go. That is the same object that rectifies a photographed façade into an elevation, used in the opposite direction — instead of moving the plane’s horizon out to infinity, it brings it in close.

The test that it is a projection #

Anamorphosis is the one place where the difference between a projection and a general distortion is easy to demonstrate, because a general distortion would do the job just as convincingly to look at.

A projection preserves two things and a warp does not: straight lines stay straight, and the cross-ratio of four collinear points is unchanged. The figure asserts both. Every straight stroke of the block letters comes out straight to arithmetic noise, and the cross-ratio along a test line agrees before and after to eight digits.

That matters because it is what makes the construction reversible and exact. A picture warped by a spline or a mesh to look right from a viewpoint would look right from that viewpoint and would not be a projection of anything: the geometry would not compose with other projections, the cross-ratios would be wrong, and a photograph of the result from the design viewpoint would not match the original.

Where the correct viewpoint is #

Every picture has a correct viewpoint. What makes anamorphosis feel different is only how far that viewpoint is from the one people naturally take.

An ordinary painting’s viewpoint is a metre or two in front of it, near the middle, at the height of the horizon. Nobody has to be told; that is where people stand.

An anamorphic picture’s viewpoint is chosen to be somewhere else — far to the side, very close to the surface, sometimes below it. The picture is exactly as correct from its point as the ordinary painting is from its. It is simply that its point is not the default, and the picture is illegible from the default.

So the two are not different kinds of picture. They are the same kind of picture with different parameters, and the reason anamorphosis is startling is that it makes the parameter visible by choosing an extreme value for it. That is the demonstration this site’s premise wants: the viewing point is part of the specification, and here is a picture where forgetting it costs everything.

Catoptric anamorphosis, which adds a mirror #

The other family puts a mirrored cylinder or cone on the surface and paints an image that resolves in the mirror.

The construction is the same idea with one extra step. Rays from the viewpoint reflect off the mirror surface and continue to the picture plane, and the image is drawn where those reflected rays land. Because a reflection is itself a projection from a reflected centre, the whole path is a composition of two projections and can be computed as one.

For a plane mirror the composition is again a homography and there is nothing new. For a cylindrical mirror the map is not a homography — the reflected rays do not form a pencil through a single point — and straight lines no longer come out straight. That is the visible signature of cylindrical anamorphosis: the painted image has strongly curved strokes even when the target image is made of straight ones.

Which is a useful diagnostic. A flat-surface anamorph has straight strokes, always. A curved-mirror anamorph does not, ever. The presence or absence of curvature says which construction was used, without knowing anything else about the picture.

What it was for #

The technique appears in Europe from the early sixteenth century and had at least three distinct uses that had nothing to do with each other.

As a demonstration. Perspective was new, contested, and being argued about as a science. An anamorph is an existence proof that the construction works: it makes a prediction — this smear will resolve into a skull from exactly there — that is falsifiable and that no other method could make. Leonardo’s anamorphic eye of around 1485 sits in a notebook among optical studies, not among finished works.

As concealment. A picture legible only from one position could carry a portrait of a proscribed monarch, a political emblem, or an image that would be indecent from the front. There is a good deal of seventeenth-century work of exactly this kind.

As a memento mori. Holbein’s skull is the famous case and its placement is doing work: from the front the painting is a display of worldly instruments — globes, lutes, books of arithmetic — and the skull is unreadable. From the position where the skull resolves, the instruments are so foreshortened as to be meaningless. The two readings are mutually exclusive by construction, which is the point being made.

The modern version #

The construction is in constant current use and is almost never called anamorphosis.

Road markings. The letters spelling BUS LANE on a road surface are elongated by a factor of four or five, so that they read correctly to a driver approaching at a shallow angle. That is a flat-surface anamorph with a viewpoint chosen from the geometry of a car.

The road case is the one with a number in it, and the number says something the practice does not advertise. A letter lying flat on the road at distance $D$, read by an eye $h$ above it, is seen at a grazing angle whose cotangent is $D/h$ — so the elongation needed to make it read correctly is exactly that ratio. At a driver’s eye height of about 1.2 m, a four-to-five-fold elongation is the correct design for a letter five or six metres ahead, and no further: at twenty metres the same letter would need to be seventeen times its width, and at fifty, forty times.

Since the elongation is linear in $D$ and the letter has to be painted once, a road legend is necessarily correct at one distance and wrong at every other — over-elongated when it is nearly underneath and under-elongated when it is far ahead. That is the anamorph’s single design eye arriving in the most ordinary place there is, and it is why painted road letters look stretched from a footbridge and never quite resolve from a car either: the driver crosses the design distance at speed and reads the word on either side of it.

Stadium advertising. The logos painted on a pitch are drawn to read correctly from the main camera position, and look grotesquely stretched from anywhere else in the ground.

Projection mapping. Projecting an image onto an irregular building requires pre-warping it so that it looks right from the audience, which is exactly the same computation carried out per-pixel on an arbitrary surface.

Augmented reality and head-up displays. Any image composited into a view of the world has to be projected for the viewer’s actual eye position, which is the anamorphic problem with the viewpoint tracked rather than chosen.

In every case the reason the technique is invisible is that it works: nobody looking at BUS LANE from a car thinks about the letters being five times too long.

Printing it, which is the only way to check it #

The claim an anamorph makes is physical: from this position, at this size, the picture resolves. That is checkable, and it is only checkable on paper at true scale with an eye where the construction says.

That is why the figure ships with the viewpoint stated as an angle rather than as a vague instruction, and why the site’s figure machinery supports a print variant at true size. A screen-sized anamorph viewed from an arbitrary distance is a picture of an anamorph rather than an anamorph.

It is also the sharpest available demonstration of the site’s whole argument, because it is the one case where being at the wrong viewpoint is not a subtle degradation but a total failure. Ordinary perspective tolerates the wrong viewing distance well — a picture seen from twice the correct distance still reads as the scene. An anamorph seen from the wrong place reads as nothing at all, and the difference between the two cases is only how extreme the viewpoint is.

Constructing one by hand #

The classical method needs no arithmetic and is worth having, because it makes clear that nothing unusual is going on.

Draw the target image on a grid of squares. Then construct the perspective image of that grid as seen from the chosen viewpoint, on the surface it will be painted on — which for a flat wall means an ordinary two-point construction with the eye far off to one side.

Now copy the image cell by cell: whatever is in square (3, 5) of the original goes into the corresponding quadrilateral of the projected grid, drawn freehand.

That is all. The distortion is carried entirely by the grid, the artist works cell by cell in an undistorted frame of reference, and the result is exact to the precision of the grid. Dürer’s woodcuts of perspective apparatus show devices for doing the same job mechanically, by tracing rays with a taut string.

The grid method also explains why anamorphic images from before the nineteenth century are usually built from simple, blocky forms: the finer the target image, the finer the grid must be, and the labour grows with the square.

The reverse operation #

Photographing an anamorph from the correct viewpoint recovers the original image, and doing so is the same homography run backwards.

That is more than a curiosity. It is the standard way of verifying that an anamorphic installation is correct: photograph it from the design viewpoint and compare with the intended image. The comparison is pixel-for-pixel, and departures show up as local displacements that can be traced back to which part of the surface is misaligned.

The same operation is what turns a photograph of a painting taken at an angle into a rectified elevation, and what a document scanner does to a photograph of a page. All three are the same map with different intentions, which is the recurring observation of this site: one operation, several uses, distinguished only by where the centre of projection is put.

What it demonstrates about ordinary pictures #

The reason anamorphosis is worth a whole essay on a site about perspective is that it is the existence proof for the site’s premise.

If a picture were simply a convincing arrangement of marks, there would be no reason for an anamorph to work at all — nothing would resolve from any particular position, because there would be no position the marks were computed for. That anamorphs work, reliably, from a predictable point, is direct evidence that a picture is a projection through a centre and that the centre is a real, locatable thing.

And it is the same centre ordinary pictures have. The only difference is that in an ordinary picture the centre is somewhere unremarkable, so nobody notices it is there, and the picture tolerates being viewed from the wrong place well enough that the question never comes up.

Anamorphosis removes the tolerance. It is the same geometry with the margin taken away, and what is left is a picture that says exactly where it must be looked at from, and says it unmistakably.

Why it stayed a curiosity #

Anamorphosis is a technique with no drawbacks except the obvious one, and the obvious one is decisive: a picture legible from a single unusual position is not much use as a picture.

That is worth stating because it explains the technique’s distribution. It appears where the single viewpoint is guaranteed by the architecture — a corridor with one entrance, a stair with one approach — and it appears where the illegibility is the content, as in the concealment cases and the memento mori. Everywhere else, a picture wants to be legible from wherever people are, which means an ordinary viewing distance and an ordinary direction.

Its modern uses are all cases where the viewpoint really is guaranteed. A road marking has a driver approaching along the road; a pitch logo has a camera in a fixed gantry; a projection mapping has an audience in a known place. The technique did not become useful when the mathematics improved. It became useful when situations arose in which the viewer’s position could be relied on.

Which is the same observation as everywhere else in this field. The geometry has been settled since the fifteenth century; what changes is whether the conditions the geometry needs happen to hold.

Checking one that has been made #

The claim an anamorph makes is testable, which is unusual for a picture, and the test is worth stating because it is the same operation run backwards.

Photograph the finished work from the design viewpoint. Compare with the image it was supposed to resolve into. Where they differ, the difference is local and maps directly back to which part of the surface is misplaced — a region drawn too far along the wall shows as that part of the image displaced in a predictable direction.

That gives a construction loop rather than a one-shot method, and it is how large installations are actually made: lay out a coarse grid, photograph, correct, refine. Each pass is a homography compared with an identity, which is a computation rather than a judgement.

The same loop verifies the viewpoint itself. If the image resolves cleanly from a position other than the intended one, the construction used a different centre from the one specified, and the discrepancy locates it. An anamorph is therefore one of the few pictures that can be debugged, and the reason is the reason this site keeps returning to: it is a projection, so it makes a prediction, and a prediction can be checked.

Two more of them, and what they add #

The expansion phase built two further anamorphs and both extend what this essay claims rather than repeating it.

The first is printable. Every viewing-distance claim on this site is conditional on an assumed display width, because the site cannot measure a reader’s screen; a sheet whose dimensions are in millimetres removes the condition. That figure ships one, at 150 × 105 mm, with three numbers on it — the height, the distance beyond the near edge, and a centre line to stand on — and it is the only claim here a reader can check with a ruler.

The second is catoptric: a mirrored cylinder standing on the sheet, with the light path bending once on its way from the paper to the eye. That one is not a homography and not a projection between planes at all. Straightness does not survive it and neither does cross-ratio, so none of this essay’s checks apply, and the check that replaces them is to run the light backwards from the paper mark and require it to reach the eye — which it does, to 1.9 × 10⁻¹⁵.

What all three share is the only thing that makes them one word. Each is a picture correct from exactly one point, arranged so that being anywhere else produces nothing at all.