The cylindrical mirror unbends it

An oblique anamorph on a flat sheet is a projection: straight strokes stay straight, cross-ratios survive, and the whole map is a homography — eight numbers. It is the same object as the rectification that flattens a façade out of a photograph, run in the opposite direction.

Put a mirrored cylinder in the middle of the sheet and none of that is true any more. The light from the paper does not travel to the eye in a straight line; it bounces once, off a curved surface, and the direction it leaves in depends on where on the cylinder it struck. Straight lines do not stay straight. Cross-ratio is not preserved. There is no matrix.

What there is instead is a construction that can be traced exactly, and a check that is better than anything the flat case offers.

What the design is, exactly #

Catoptric anamorphosis is usually described loosely: the smear on the paper “turns into” a picture in the mirror. That is not a specification, and without one there is nothing to compute.

The specification used here is the tightest available. The intended picture is the one that appears painted on the cylinder’s own surface, at azimuth φ around it and height t up it. That is a well-defined object — a picture wrapped on a cylinder, which unrolls to an ordinary flat picture — and it is what the reader sees when they look into the mirror, because what they see at a point of the mirror surface is a direction, and the direction is all a single-viewpoint picture is.

Given that, the construction is a ray trace and it is four lines.

Take the point on the cylinder at (φ, t): C = (R sin φ, t, R cos φ). Its outward normal is n = (sin φ, 0, cos φ). Take the ray from the eye E to C, direction u. Reflect it: r = u − 2(u·n)n. Follow r down from C until it meets the paper at y = 0. That is where the mark goes.

Three of the steps can refuse, and all three are refusals rather than fudges. A point on the far side of the cylinder — where the eye cannot see the surface at all — has no light path and returns nothing. A reflected ray that leaves upward never lands and returns nothing. A ray that lands underneath the mirror’s own footprint is invisible from the eye and returns nothing.

The check runs the light backwards #

The flat anamorph’s check is that a camera at the eye sees the marks covering the picture. That check is available here too and it is weaker than it looks, because it is nearly a tautology: the marks were constructed on rays through the eye, so of course they lie on rays through the eye.

The check that has content is the reflection law, and it is checked by reversing the light.

From the paper mark, aim at the mirror point it came from. Reflect that direction in the cylinder’s normal. Follow the reflected ray to the eye’s height, and measure how far it misses the eye. The miss is 1.9 × 10⁻¹⁵ units on a mirror of radius 1.

A sign error in the reflection formula — the difference between u − 2(u·n)n and u + 2(u·n)n, or a normal pointing inward instead of outward — produces a smear that looks exactly like an anamorph. It has the right general character: crowded near the mirror, spreading outward, curved around the footprint. It turns back into nothing at all, and only running the light backwards says so.

That is the same reasoning as the site’s mirror essay, where a reflection computed two ways disagreed by 906 pixels and agreed to 6 × 10⁻¹⁴ once one image axis was reversed. Two routes to one answer, and the disagreement is the measurement.

The half of the cylinder that does not exist #

One of the three refusals in the construction is worth a section of its own, because it decides the shape of every catoptric anamorph ever drawn.

Only the half of the cylinder facing the eye reflects anything the eye can receive. The far half is there, and it is mirrored, and it is invisible: a ray from the eye toward a point on the far surface has already passed through the near surface, and the physical object it would strike is the inside of an opaque tube.

The construction refuses those points rather than computing a mark for them. That is not tidiness — the reflection formula happily produces a direction for a point on the far side, and the ray happily meets the paper somewhere, and the mark drawn there would be a perfectly plausible-looking part of the smear that reflects nothing.

The consequence is that a catoptric design can occupy at most half the cylinder’s circumference, and in practice rather less, because the reflection at grazing incidence near the silhouette edges is useless. The design here spans about 86° of azimuth, and the historical examples are in the same range.

That in turn sets the shape of the paper. A design confined to a band of azimuth reflects into a fan on the sheet, opening away from the eye, and the fan’s angular width is roughly the design’s own. Every classical example has that fan, and every one has the mirror standing on an otherwise blank disc of paper — the region under and immediately around the cylinder, where a reflected ray cannot land because the mirror is in the way.

The blank disc is the third refusal, and it is the one a reader can verify without any arithmetic: the smear never touches the mirror’s footprint. If it did, that part of the design would be hidden by the very object meant to reveal it.

How uneven the map is #

The first attempt to measure “how anamorphic is this” compared the bounding boxes: the aspect ratio of the smear against the aspect ratio of the wrapped picture. It reported 2.34 against 2.42, which says the two are nearly the same shape.

They are visibly not. The smear is unreadable and the picture is a word.

Aspect is the wrong statistic and the reason is the same one the flat anamorph essay reaches. A drawing is unreadable because its scale changes across it, not because the whole thing is elongated. A uniform stretch by any factor is undone by the eye without effort; a stretch that varies is not.

So the measure used is the local area scale: take a small patch of the intended picture, map it through the mirror to the paper, and compare the two areas. Sample that across the design and report the ratio of the largest to the smallest.

It is 7.2. One part of the design gets seven times as much paper per unit of picture as another, and that is what makes it unreadable flat. A map with that ratio near 1 would be a similarity — the same picture, moved and scaled — which is to say not an anamorph at all.

The assertion in the site’s gate is written on that number rather than on the bounding boxes, and the bounding-box version is recorded here rather than deleted, because a measurement that reported 2.34 against 2.42 for a design that is obviously wildly distorted is a measurement worth remembering not to trust.

Why the mirror is the right shape #

A cylinder is a peculiar mirror to choose and the choice is not arbitrary.

Its axis is vertical, so it does nothing to the vertical structure of the design that depends on azimuth: a point at azimuth φ reflects to a paper mark at an azimuth determined by φ and by the eye, and the map from φ to that azimuth is the same at every height. The mirror separates the two coordinates almost completely.

That is why a catoptric anamorph is drawn as a fan around a circle rather than as an arbitrary blob. The angular coordinate of the design maps to an angular coordinate on the paper, monotonically; the height coordinate maps to a radius. The design is in polar coordinates about the mirror’s foot, which is why every historical example — and there are a great many, mostly seventeenth- and eighteenth-century, mostly on playing cards and snuff boxes — has the same distinctive radial character.

A conical mirror does something different and produces the other classical form: the design becomes a ring, and the picture appears looking down into the cone. A pyramidal mirror gives four pictures. The family is large and each member is one ray trace away.

What none of them are is a projection onto a curved picture surface, which is the subject of this site’s curved field. A fisheye and a cylindrical panorama are maps from directions to marks, with the eye at the centre of the sphere of directions. A catoptric anamorph is a map from a picture to a picture, with a reflection in the middle. The two are easy to confuse and behave nothing alike.

What the reader actually has to do #

The practical arrangement is worth stating because it is the point at which this stops being geometry.

A mirrored cylinder of the stated radius stands at the centre of the sheet. The reader’s eye goes at the stated height above the paper and the stated distance from the mirror’s axis. Both numbers are in the design, and both matter — the same argument as everywhere else on this site, in its strongest form, because the failure from the wrong place is total.

In practice a catoptric anamorph is much more forgiving of the distance than of the height. Moving the eye further back changes the reflected geometry slowly, because the rays from a distant eye are nearly parallel and the reflection off a given point of the cylinder barely changes. Moving the eye up or down changes which band of the cylinder the paper reflects into, and the picture slides and squashes immediately.

That asymmetry is a consequence of the mirror’s shape rather than of anything about anamorphs, and it is the reason the historical examples come with a cylinder of a specified diameter and no instruction about where to put a face.

The three anamorphs, and what they share #

This site now has three constructions under the same word and they are worth setting side by side, because the differences are more instructive than the similarity.

The homographic anamorph is a projection of a picture plane to a picture plane. Straight stays straight, cross-ratio survives, eight numbers, and it is the same object as a rectification.

The shadow anamorph is a projection from the eye onto the sheet, of a picture standing upright on it. It is the same operation as a shadow with the lamp moved to the eye, and it is the homographic one in a form where the construction is visible rather than the matrix.

The catoptric anamorph is a ray trace with one reflection. Nothing survives: not straightness, not cross-ratio, not any invariant the rest of this site runs on.

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 at any other point produces nothing. That is the site’s premise with the tolerance turned down to zero. An ordinary picture viewed from the wrong distance is a correct picture of a different scene, and nobody notices. An anamorph viewed from the wrong place is a smear, and everybody does.

The mirror is the version where even the light path is not straight, and it still comes back — which is worth having, because it shows that what the premise depends on is the uniqueness of the viewpoint and not the simplicity of the geometry.

Why this one is not printable #

The flat anamorph in the previous essay ships a sheet in millimetres with three numbers on it, because paper is the one surface whose width the site knows. This one does not, and the reason is worth stating rather than leaving as an omission.

A catoptric anamorph needs a physical object of a stated radius. The design is computed for a mirror of radius R with the eye at a stated height and distance in units of R, and printing it at any size at all is useless without a cylinder that matches. A reader with a printer has a sheet; a reader with a printer and a mirrored tube of unknown diameter has a sheet and a disappointment.

That is a different kind of dependency from the flat case and it is worth being precise about which. The flat sheet’s claim is conditional on the print scale, which the file can fix. The mirror sheet’s claim is conditional on the print scale and on an object the file cannot supply, and the two have to agree with each other to within a few per cent or the design lands on the wrong band of the cylinder.

The honest options are to specify a common object — the classical examples assume a particular size of polished tube, and modern reproductions usually ship one — or to say plainly that this figure is a computation shown on a screen rather than an artefact. This site takes the second, because shipping a sheet that requires an unstated object would be exactly the kind of silently-conditional claim the whole site exists to argue against.

Why these were made on playing cards #

The historical record of catoptric anamorphosis is not in treatises. It is on small portable objects — playing cards, snuff-box lids, fan leaves, printed sheets sold with a tube of polished metal — mostly from the seventeenth and eighteenth centuries, and mostly carrying subjects that would have been awkward to display openly.

The physical reason for the format is in the geometry. The design has to sit within a fan of a few dozen degrees around a mirror of a specific radius, and it has to be looked at from a specific height above the sheet. A large picture cannot be made this way: doubling the design’s size doubles the mirror, and a mirrored cylinder the size of a column is not something anybody holds. The construction has a natural scale, set by the size of an object a hand can hold and an eye can lean over, and that scale is a playing card.

There is a second reason, and it is the interesting one. The smear is not a picture of anything until the mirror is present, and the mirror is a separate object. A sheet on its own carries no image; a sheet plus a tube carries one. That is a property no other picture-making method has, and it makes the technique a way of storing an image in two parts, either of which is innocuous alone.

Both reasons come out of the same measurement — the scale varying by a factor of seven across the design. It is what makes the sheet unreadable, what makes the mirror necessary, and what confines the whole family to objects small enough to carry.

What is not claimed #

Two boundaries, stated rather than implied.

The reader looking into the mirror sees the reflection in the direction of the cylinder’s surface, not at the depth of it. A single-viewpoint picture is a set of directions and nothing else, so that is enough for the construction to be correct — but a real reader has two eyes, and the two receive slightly different reflections, which supplies exactly the information that the picture is a reflection of a flat sheet. Catoptric anamorphs read better with one eye closed, and this is why.

And the mirror is treated as a perfect cylinder of zero thickness with a perfectly reflecting outer surface. A real mirrored tube reflects at its outer surface and again, faintly, at the inner one; a real polished cylinder is not exactly circular; a real reader’s eye is not a point. All three are reasons the printed result is approximate, and none of them is a reason the geometry is.