A map along, and a picture across

The parallel field on this site rests on one measurement. A parallel projection sends the midpoint of a world segment to the midpoint of its image; a perspective projection does not; and that is the exact difference between the two families, which is why neither is a defective version of the other. A system that preserves measure cannot show convergence and a system that shows convergence cannot preserve measure.

Run that same test on a scroll and it gives two different answers depending on which way the segment points.

The two answers are not approximate #

The along-the-roll answer is zero, not small. The map from world $x$ to paper $u$ is $u = u_0 + sX$, a scaling, and a scaling preserves every affine relationship there is: midpoints, ratios of division, parallelism, the lot. There is nothing to approximate.

The across-the-roll answer is the ordinary perspective one. The map from $(Y, Z)$ to $v$ is a projective divide, so the midpoint drifts, and the drift grows with the depth range exactly as it does in any perspective picture.

Two exact answers, from one projection, in two directions. No other system on this site does that. The parallel systems answer preserved in every direction; a pinhole answers destroyed in every direction; a curved picture surface answers destroyed in every direction and in a different amount per direction, which is anisotropy of degree rather than of kind. A scroll answers differently in kind.

What that buys along the roll #

The consequence is best stated as what a reader holding a ruler can recover, and here it is worth being precise, because the answer is stronger than perspective ever gives.

Distances along the roll are recoverable up to one global scale. Two courtyards measured on the silk are in the same ratio as two courtyards on the ground. A reader who knows the width of any one of them knows the width of every one, and the length of the whole journey, and the spacing of everything along it — with no vanishing point, no reference to the horizon, no cross-ratio, and no assumption about the camera.

Compare what a perspective picture gives along the same direction. To get a ratio out of a photograph of a receding row, the standard route needs a cross-ratio with the vanishing point as the fourth point, and that requires the vanishing point to be findable, the four points to be collinear, and the marks to be accurate — and it returns a ratio, not a length, so a reference is still needed. The scroll skips all of it. Along the roll a scroll is a scale drawing, and a scale drawing is a strictly stronger thing to have than a projective picture.

And what it costs across it #

The other direction is a perspective picture with all of the usual costs and one extra.

The usual costs are the ones the whole site is about: length, angle, area and the ratio of lengths all go, and the cross-ratio survives. The extra cost is that in a scroll the cross-ratio does not survive either, because four collinear world points do not image to a line at all — the image of a straight line is a hyperbola, so there is no image line to compute a cross-ratio on.

So the across-roll direction is worse than a perspective picture, not equal to one. It has the depth compression without the invariant that normally compensates for it. Anything a reader might want to recover about depth from a handscroll has to come from the column rather than from any relationship between columns, and a column is one vertical line.

That is a genuinely unusual place in the space of drawing systems: exact measure in one direction and less than perspective’s usual consolation in the other. It is also, on reflection, exactly matched to what a handscroll is for. A journey along a river is a sequence of places at known spacings; the depth of any one of them is scenery.

The measurement, and what makes it a measurement #

scrollRatioSurvival runs midpointDrift — lib/scene.js’s function, imported rather than rewritten — on two segments, one along the roll and one across it. The import matters for the same reason the ray solver’s did in the previous essay: this is the site’s existing definition of does this projection preserve ratio, and answering it a second way here would make the scroll’s answer incomparable with the six parallel systems’ answers, which is the comparison the whole field is building toward.

The numbers: 0 px of drift along the roll, and 21.9% of the drawn segment across it, for a segment running from 2 m to 16 m of depth. The first is a control in the strict sense — it is what the function returns for a projection that preserves ratio, and the parallel systems return the same — and the second is the measurement.

Both are asserted rather than reported, and the assertion on the second has a floor rather than a ceiling: it requires the drift to be large, above 2% of the segment. An assertion that a number is small is satisfied by a broken computation returning zero; an assertion that a number is large is not.

Two things a scroll can be measured for, and one it cannot #

It is worth working the recovery question through properly, because “a map along and a picture across” is a slogan until it is turned into a list of what a reader can actually get.

Spacing along the journey: available, exactly, up to one scale. Any two intervals along the roll are in the true ratio of the intervals on the ground. This holds at every depth simultaneously, which is the part with no analogue anywhere else on this site: a courtyard in the foreground and a hill three kilometres back are measured by the same ruler on the same silk. In a perspective picture that comparison needs the depths of both, and the depths are exactly what a single view does not carry.

Heights within a column: available, on the usual perspective terms. A column is a one-dimensional perspective view with a real focal length, so a vertical in it is foreshortened by depth in the ordinary way, and recovering a height needs a reference at the same depth — the same condition the height recovery states for a photograph.

Depth: not available at all. Not “available with a reference”, not “available up to scale” — absent. Depth in a perspective picture is recoverable because it is encoded in the convergence of parallels, and there is nothing to converge here: along the roll the projection is parallel, so parallels stay parallel, and across the roll there is only one column’s worth of information at any position. A scroll’s picture of a courtyard does not say how deep the courtyard is, and no amount of care with the marks changes that.

That third item is the honest cost and it is a large one. A handscroll is a scale drawing of a line through a landscape, with pictorial information hung on it, and the pictorial information is not metric. Which is a fair description of what handscrolls depict.

Why the drift across the roll is the ordinary number #

One check worth making explicit, because it is the kind of thing that would otherwise pass unexamined: the 21.9% drift across the roll should be exactly what a perspective camera of the same focal length and standoff would give for the same segment, because the across-roll map is that camera. If it were not, the model would have something extra in it.

It is. The across-roll formula is $v = v_0 + f(h_e - Y)/(Z - z_c)$, which is the pinhole’s vertical coordinate with the pinhole’s focal length. So the midpoint drift for a receding segment is the pinhole’s midpoint drift, and the two libraries agree because they are computing the same expression.

That is worth having because it locates the scroll precisely in the space of systems rather than vaguely between two of them. A scroll is not between parallel and perspective, and it is not a blend of the two. It is the product of one of each — a parallel projection along one axis multiplied by a perspective projection along the other — and every property it has is the corresponding property of one of the two factors. Nothing about it is a compromise, and nothing about it is new.

That framing also says what else is in the family. A camera whose both axes are parallel is an orthographic projection; whose both axes are perspective is a pinhole; one of each is a pushbroom. The fourth combination — perspective along the roll and parallel across it — is a legitimate object nobody has any use for, and it would draw a scene that recedes sideways and does not recede in depth.

The direction that gets which #

One small observation before the framing, because it is the kind of thing that is obvious once stated and invisible before.

Which of a scroll’s two directions gets the exact measure is not a choice. The roll direction gets it because the roll direction is the one the eye travels along, and the eye travelling is what makes that map a scaling. A painter could not have assigned the property to the other axis without changing what a scroll physically is.

So the anisotropy is not a stylistic decision applied to a neutral geometry. It is the geometry of the object’s own physical form, and the drawing system is the shape of the paper made into a projection.

Anisotropy is not distortion## Anisotropy is not distortion #

There is a word that wants to be used here and should be resisted. A system that behaves differently in two directions sounds like a system that distorts, and distortion is a word for a departure from something correct.

Nothing here departs from anything. The map from world to paper is a rule, applied consistently everywhere on the picture, whose properties happen to differ between two directions. A Mercator projection has the same character — conformal everywhere and area-preserving nowhere — and nobody calls its behaviour a distortion of a globe, because a globe is not a page and something has to give.

What is worth saying instead is the trade, stated as a trade:

A scroll gives up the single centre and buys exact measure along one axis. Not approximate measure, not measure-after-rectification, not measure-given-a-reference — exact ratio, everywhere on the roll, at every depth simultaneously.

A pinhole gives up all measure and buys one point from which the whole picture is a correct projection of the scene. That point is worth having: it is what makes a photograph evidence, what makes a height recoverable, and what makes two views enough to reconstruct a scene.

Neither is a better answer, and the fifth field on this site exists to say so with a table rather than an adjective.

The direction nobody chose, and the one that was #

A last observation, and it is about which of the two directions is which.

A handscroll’s roll direction is horizontal, and horizontal is the direction along which a journey, a procession, a river and a city are extended. The direction that gets exact measure is the direction the subject is extended in, and the direction that gets ordinary perspective is the one it is not. That looks like a fit between convention and subject and probably is one — but it is worth noticing that the fit is forced by the object rather than chosen. A scroll unrolls one way. Whatever direction that is, it is the one that gets the scaling, because the scaling is the unrolling.

So the system is, in a fairly precise sense, the geometry of its own physical form. The paper’s long axis is a distance axis because the paper advances; the short axis is a picture because a picture is what fits on it. That is a stronger relationship between a medium and its geometry than perspective has with the panel — a panel could be any shape and the projection does not care — and it is the sort of thing that is invisible until the map is written down.

The two neighbours of this result #