An anamorph at true size, on paper

Every figure on this site states the distance it is correct from, and every one of those statements carries a condition: shown 160 mm wide. The site cannot measure the reader’s screen. A figure that is 200 mm wide on a laptop and 70 mm wide on a phone has two different correct viewing distances, and all the arithmetic can do is state the assumption and give the reader the means to redo it.

There is exactly one way out of that, and it is paper. An SVG whose width and height carry millimetre units and whose coordinates are in the same units prints at a known size. Print it at 100%, and the viewing distance stops being conditional.

This essay ships one. It is an anamorph, because an anamorph is the case where the claim about viewing position is strongest — a picture that is unreadable from anywhere except one place, so that being in the wrong place is not a subtle degradation but a total failure.

The construction is a shadow #

An anamorph is usually introduced as a distortion applied to a picture, with a homography or a grid or a stretching. That description is correct and it hides what is happening.

Here is the whole construction. Stand the intended picture upright, on the sheet. Put an eye somewhere low and off to one side. From the eye, project every point of the upright picture down onto the sheet — a straight line from the eye through the point, continued until it meets the paper. Mark where it lands.

That is projectToGround(centre, point), and this site already has it, because it is the function that computes a shadow. Put a lamp at the centre and the output is the shadow of the object on the ground. Put the eye at the centre and the output is an anamorph.

So: an anamorph is the shadow of the intended picture, cast from the eye that is meant to read it. Not analogous to a shadow, not a kind of shadow — the same function, the same arguments, the same output.

Why it works follows in one line. A point of the upright word and its mark on the paper lie on the same ray from the eye, because that is how the mark was constructed. Two points on one ray are seen in the same direction. So from that eye, and only that eye, the marks cover the letters exactly.

The check is that arrangement run backwards. A camera is put at the eye position and both the upright word and its flattened shadow are projected. The worst disagreement over every corner of every glyph is zero to the last bit — which it must be, since the two are on the same rays, and which would not be if the centre of projection or the plane had been got wrong.

The one machine, a third time #

This site has a thread for constructions that turn out to be one operation used more than once, and this is its third instance.

A shadow is a projection from the lamp onto the ground. A reflection is the view from a camera on the far side of the mirror. An anamorph is a projection from the eye onto the paper. Three subjects, taught in three chapters of three different books, and one function with the centre moved.

The value of noticing is not tidiness. It is that a function checked once is checked for all three, and the checks transfer. The shadow essay establishes that the shadow of a straight line is straight, by sampling and measuring the bend — 8.6 × 10⁻¹⁴ px over 294 px. That result applies to this construction unchanged, and it is what guarantees that a straight stroke of the upright word stays a straight stroke on the paper.

That property is what makes an anamorph a projection and not a warp. A warp bends things. A projection sends straight to straight and preserves the cross-ratio along every line, and both are checked here rather than assumed.

What the sheet says #

The printable sheet carries one line of text and the line is the point of the whole exercise:

Print at 100%. Put one eye 52 mm above the sheet, 80 mm beyond the near edge, on the dashed centre line, and read.

Three numbers and a dashed line. No assumed display width, no clause beginning “if the screen is about”, no arithmetic left to the reader. The sheet is 150 mm wide because it says it is 150 mm wide, and a printer that honours the units makes it so.

That is a stronger claim than this site makes anywhere else, and it is worth being clear about why it can be made here and not elsewhere. The screen figures are correct projections whose viewing distance depends on a width the site cannot measure. The sheet is a correct projection whose viewing distance depends on a width the sheet itself fixes. The difference is not in the geometry — it is in whether the physical scale is known.

What being in the wrong place costs #

The reason to use an anamorph for this demonstration rather than an ordinary perspective is that an ordinary perspective is forgiving and an anamorph is not.

A normal picture viewed from twice its correct distance is still a correct picture — of a different scene, with its depth compressed. The viewer sees a room that is shallower than the one depicted and does not notice, because there is no comparison available and the visual system is not measuring. This site has said so repeatedly, and it is why five centuries of pictures were made and hung without anybody minding where they were looked at from.

An anamorph viewed from the wrong place is not a picture of anything. The letters are stretched by a factor that varies across the sheet — here they run two and a half times as deep as they are tall — and no reading of them is available from anywhere except the one point. The failure is complete rather than graded, which makes the claim checkable by a reader with a printer in a way that a claim about a shallow room is not.

The drag on the figure makes the dependence visible. At an eye 38 mm above the sheet the marks run 4.4 times as deep as the letters are tall; at 100 mm they run 1.0 times, which is barely an anamorph at all. Raise the eye further and the marks contract toward the word’s own footprint, until in the limit of an eye directly overhead the whole word collapses onto the line it stands on — the shadow of an upright thing under an overhead light.

So the construction fails at both ends and in different ways. Too high and there is nothing to decode; too low and the marks run off the sheet, which is why the figure asserts that the whole anamorph lands on the paper, front to back and side to side, and refuses to draw one that does not. An anamorph half of which is off the page is not a weaker anamorph, it is not one.

Why straight strokes matter #

The glyphs on the sheet are drawn as closed polygons with straight edges, and not with a font. That is deliberate and it is not a rendering convenience.

A font’s outlines are curves. Curves do not have the property this figure is about: the image of a Bézier under a projection is another curve, but “the curve stayed a curve” is not a checkable statement in the way “the straight line stayed straight” is. Block letters made of straight strokes turn the whole claim into a set of collinearity tests, and every one of them is available to the machinery already here.

So the letters are built from rectangles and parallelograms, and the check is that every stroke’s edge is still a straight line after the projection, and that the cross-ratio of four points along it is unchanged. If the projection had a bug — a wrong plane, a wrong centre, a perspective divide in the wrong place — the marks would still look like a plausible anamorphic smear. Straightness and cross-ratio are what notice.

That is the same reasoning as the geometry being drawn from a camera rather than constructed by eye. A picture no projection could produce should fail to build, not appear with nothing to say so.

The stretch is not uniform, and that is the whole trick #

A reader who has followed the shadow construction may reasonably ask why the result is unreadable at all. If the marks are the word stretched away from the viewer, why is stretching it back not obvious?

The answer is that the stretch is not a single factor. Every point of the upright word is projected with its own multiplier, t = e/(e − h), where h is that point’s height above the sheet and e is the eye’s. The top of a letter is stretched by 1.55 and its base by exactly 1 — the base is already on the paper and does not move at all.

So the marks are not the word scaled; they are the word scaled by a different amount at every height, with the multiplier growing without bound as the height approaches the eye’s. A letter’s vertical stroke becomes a wedge. A horizontal stroke at the top of the word becomes much wider than the same stroke at the bottom. No single affine correction undoes it, and the eye, which is very good at compensating for uniform scaling and shear, has nothing to apply.

This is the same quantity the mirror anamorph reports as a ratio of local area scales, and it is the honest measure of “how anamorphic is it”. Two drawings can have the same overall proportions and be wildly different in readability, depending on whether the scale varies across them. The bounding box says nothing; the derivative says everything.

It also explains the one direction from which an anamorph is nearly readable and is not the correct one. Standing at the right height but too far back compresses everything toward the correct picture without getting there, so the word becomes legible-ish and wrong — which is why the sheet gives three numbers rather than two.

Where the eye has to be, in three numbers #

The sheet gives a height, a distance beyond the near edge, and a line to stand on. All three are necessary and it is worth saying which failure each one prevents.

The line fixes the left-right position. Off it, the whole anamorph is seen obliquely from the side, and the word shears — the strokes stop being vertical and the letters lean. This is the failure a reader will notice first and correct instinctively, because a leaning word looks wrong in a familiar way.

The height fixes the vertical multiplier. Too high and the letters are squat; too low and they are drawn out. This is the failure the drag on the figure demonstrates, and it is the one that varies fastest.

The distance fixes the rest. Standing at the right height but further back flattens the projection toward a parallel one, and the word becomes almost readable and systematically wrong in its proportions — which is the most misleading of the three, because it produces something legible enough to be accepted.

Three numbers is the minimum, because the eye has three coordinates and the construction used all three of them. Any anamorph instruction that gives fewer is leaving one of them to be guessed, and the guess will be the distance, because it is the one that fails most gently.

The limit of the sheet #

Two honest caveats belong on the printable version and are stated here rather than in small print.

The first is that a printer may not honour the units. Most will print an SVG with width="150mm" at 150 mm; some will scale to fit the page, and a browser’s print dialogue will happily apply a “shrink to fit” that ruins the whole point. The sheet has a dashed centre line, which gives a reader something to measure — if it is not exactly 105 mm long, the print was scaled and the stated eye position is wrong by the same factor.

The second is that a human eye is not a pinhole. The construction places a point; a reader has a pupil several millimetres across, two of them, and a visual system that will fuse what it is given. In practice the anamorph reads over a region a centimetre or two wide rather than at a mathematical point, and it reads best with one eye closed — because binocular disparity supplies the information that the marks are on a flat sheet, which is exactly the information the illusion needs suppressed.

That second caveat is a fact about seeing rather than about projection, and this site’s habit is to keep the two apart. The geometry says where the eye must be for the marks to lie on the letters’ rays. What a reader actually perceives when they get there is a different question, with a different kind of answer, and the sheet’s claim is only the first of the two.

What a printed figure would change #

The sheet in this essay is one figure, and it is worth asking what it would mean for the rest of the site if every figure had one.

Every viewing-distance statement here has the same shape: shown 160 mm wide, this picture is correct from 22 cm. The 160 is a stated assumption and the 22 follows from it exactly. A reader on a phone can redo the arithmetic — the ratio is printed, and multiplying it by their own measured width gives their own distance — and almost nobody will.

A printed version removes the step. The sheet is 150 mm because the file says 150 mm, so the distance is 52 mm because the file says 52 mm, and there is nothing to redo. The claim goes from conditional to checkable with a ruler.

The reason the whole site is not built that way is that most of its figures are not the kind of thing anybody prints. A plot of edge stretch against field of view has a correct viewing distance in the same sense every picture does, and the number means nothing for it, because a plot is a diagram of a relationship rather than a projection of a scene. The viewing-distance claim only has content for figures that are projections — the room, the box, the pavement, the anamorph — and of those, the ones worth carrying to a printer are the ones where being in the wrong place is fatal.

Which is one figure. That is a small return, and it is the honest size of it: paper removes the site’s only unmeasurable assumption, for the one claim where the assumption mattered most.

The next centre #

Moving the centre of projection is what generated all three constructions in this essay’s second section, and there is one more place to move it that changes the character of the result entirely: put a mirror in the path.

With a curved mirror standing on the sheet, the ray from the eye no longer travels in a straight line to the paper — it reflects, and the map from the intended picture to the marks stops being a homography and stops being a projection in the plane sense at all. The intended picture can then be wrapped around a cylinder, and what stands up in the mirror is a picture that no flat construction could have produced.

That is the classical catoptric anamorph, computable exactly by tracing the reflection, and it is the subject of the next essay.