A pond in a scroll is not an ellipse

A perspective picture of a circle is a conic, always. That is one of the few statements about perspective that holds without exception, it is the foundation of every construction for drawing a round thing in a picture, and it comes with a well-known trap: the drawn ellipse’s centre is not the image of the circle’s centre, so the circle’s centre moves and a draughtsman who places a wheel’s axle at the middle of its drawn ellipse has placed it wrongly.

A handscroll is not a perspective picture. A scroll is a camera that moves: orthographic along the roll, perspective across it. A straight line in a scroll is a hyperbola found that the first casualty of that geometry is straightness, and a map along, and a picture across found that the along-roll direction keeps every ratio a pinhole destroys. Both of those are statements about lines. A garden in a handscroll has ponds, and a pond is a circle lying on the ground.

So the question is concrete: what does a scroll draw a round pond as?

The equation a scroll’s pond lies on #

Write the scroll as the earlier essays wrote it. The eye travels along a track at height h, standing off a distance c from the ground’s origin, and draws one vertical column at a time. A world point at (X, Y, Z) lands on the paper at column $u = sX$, a scaling along the roll with compression s, and at row $v = v_{0} + f(h - Y)/(Z + c)$, the pinhole’s own vertical coordinate across it.

A pond of radius R centred at $(x_{0}, 0, z_{0})$ is every ground point with $(X - x_{0})^{2} + (Z - z_{0})^{2} = R^{2}$. Put the paper coordinates in: $X = u/s$, and writing $w = v - v_{0}$, $Z + c = fh/w$. Substituting and clearing the denominator gives

$$\left(\frac{u}{s} - x_{0}\right)^{2} w^{2} + (fh - Dw)^{2} - R^{2}w^{2} = 0, \qquad D = z_{0} + c,$$

which has a $u^{2}w^{2}$ term. A conic has terms of degree two and no higher; this curve has one of degree four. The pond lies on a quartic.

The figure checks the equation against the drawing rather than trusting the algebra. Three hundred and sixty points of the pond are projected by the scroll, and every one of them satisfies the quartic, divided by $(fh)^{2}$ so the residual is a pure number, to 2 × 10⁻¹⁶. The least-squares conic used to find the ellipse in every imaged circle is then fitted to the same points, and the drawing departs from that best conic by up to 3.08 px.

Beside it, for control, is the pinhole that has exactly the scroll’s columns: the same eye height, the same standoff, the same focal length across the roll, looking straight ahead. Its rows are the scroll’s rows to the last bit, so the two drawings differ only in the along-roll coordinate. Its pond is a conic to 1 × 10⁻¹² px. The conic fit is working, and the scroll’s pond is not one.

The scroll’s pond is the pinhole’s, stretched row by row #

There is a way to see the quartic without the algebra, and it explains every other result in this essay at once.

The scroll and the pinhole of its columns draw every point on the same row. They differ only in the column. The pinhole puts a point at column $fX/(Z + c)$; the scroll puts it at $sX$. Divide one by the other and the ratio is $s(Z + c)/f$, which depends only on the point’s depth — and on a shared row the depth is fixed, since $Z + c = fh/w$. So on every row the scroll’s drawing is the pinhole’s drawing stretched sideways by a factor proportional to $1/w$: far rows, close to the horizon, stretched a great deal, and near rows stretched little.

The scroll’s pond is therefore the pinhole’s ellipse with each row stretched in inverse proportion to its height below the horizon. Stretching an ellipse by a single factor keeps it an ellipse. Stretching it by a factor that varies from row to row bends its outline into a curve of higher degree, and $1/w$ is exactly the variation that produces the $u^{2}w^{2}$ term.

It also predicts the direction of the departure. The rows nearest the horizon — the far shore of the pond — are stretched most, so the scroll’s pond is broader across its far half, and narrower across its near half, than any ellipse matching its overall width and depth. A best-fitting conic splits the difference and misses both halves, and that is where the 3.08 px lives.

The row where the pond is widest #

The quartic is the negative result. The positive one is in the dashed lines, and it is the more surprising of the two.

In the pinhole’s drawing, the widest row of the pond — the row joining its leftmost and rightmost points — lies 3.58 px from the row on which the pond’s centre is drawn. That is the familiar error. The widest horizontal chord of the drawn ellipse is not the image of the circle’s widest chord, because the pinhole divides the across-picture coordinate by depth too, and the extreme left and right of the drawing come from points of the circle slightly nearer the eye than its centre, where the division makes them wider.

In the scroll’s drawing the widest row lies on the centre’s row exactly: 0 px apart, to the arithmetic floor.

The reason is the along-roll scaling. A scroll draws a ground point’s column as its world position along the track, multiplied by a constant, whatever its depth. So the widest row of the drawn pond is the row of the circle’s widest extent along the track — its diameter parallel to the track — and every point of that diameter lies at the depth of the circle’s centre. The row that depth is drawn on is the centre’s row. There is nothing to correct and nothing to warn a draughtsman about: the scroll puts the pond’s widest reach exactly where its middle is.

So the two drawings have traded. The pinhole keeps the conic and loses the row; the scroll keeps the row and loses the conic. It is the same trade a map along, and a picture across found for midpoints, seen on a curve: along the roll a scroll is a scale drawing, and a scale drawing puts a circle’s widest chord at its centre.

Off to one side, the pinhole’s pond leans #

Move the pond four metres to one side of the eye’s line and a second difference appears.

The pinhole’s pond now leans: its leftmost point and its rightmost point are drawn 9.67 px apart in height, because the side of the pond nearer the eye’s line and the side further from it are at different angles to the line of sight. This is the same effect which way the drawn circle leans measured for a circle in an ordinary perspective drawing: an ellipse drawn off the centre of a picture tilts, and the taught rule for which way is right only on the principal ray.

The scroll’s pond, four metres to the side, is the identical shape it was straight ahead, widest on its centre’s row and not leaning at all. That also follows from the scaling. Moving a pond along the track moves every point of its drawing by the same number of columns and changes no row, so the drawing is translated and nothing else. A scroll has no side. Every column is straight ahead of the place the eye was when it drew that column, and a pond at the far left of a scroll is drawn exactly as a pond at its centre.

That property has a plain consequence for a reader of a handscroll: shapes along its length can be compared directly. A round pond near the beginning of the journey and one near the end, at the same depth, are drawn as the same shape at the same size. In a panorama or a wide perspective picture that comparison needs a correction that depends on where each pond sits in the frame.

A width that is a measurement, and a height that is not #

The two drawings have one more thing to trade, and it is what a reader could recover from each.

The scroll’s pond is 156.0 px wide on the paper. That is $2Rs$ exactly — twice the three-metre radius times the scroll’s 26 px per metre along the roll — with no dependence on how far out the pond is. A reader who knows the roll’s scale reads the pond’s diameter straight off its width, to the precision of the drawing, and would read the same number for the same pond drawn at twelve metres. The pinhole’s pond is 222 px wide at five metres and narrower further out; its width is a measurement only together with a depth.

Height goes the other way, and it goes the same way for both drawings. They share every row, so the scroll’s pond is exactly as tall as the pinhole’s, and it inherits the pinhole’s error in that direction. The pond’s centre is drawn 3.70 px from the row halfway between its nearest and farthest points, in both drawings, because across the roll the scroll is the pinhole. A draughtsman who placed the pond’s middle halfway down its drawn height would be wrong by the same amount in either system.

So the scroll’s pond carries the same division of labour a map along, and a picture across found for segments: exact in its width, perspective in its depth. It is the widest row that makes the result look surprising, and it stops looking surprising once the width is seen for what it is — a scale drawing’s width, on the row where the circle’s widest chord lies. The centre a scroll does not have is a missing common point of rays; it is not a missing centre of shapes, and a pond is where the difference shows.

Three pixels on a thirty-pixel pond #

Whether a departure of 3.08 px matters depends on what it is measured against, and the obvious comparison is misleading.

Against the pond’s width of 156 px, three pixels is two per cent, and an ellipse two per cent out of shape would pass unnoticed. But a pond on the ground is seen at a grazing angle, so its drawing is far wider than it is deep: the scroll draws this one 156 px across and only 29.6 px from near shore to far. The best conic’s miss is a tenth of the pond’s drawn depth. A shape a tenth wrong across its short dimension is not a subtle departure; it is the difference between an oval and a lens-shaped pond whose far shore is visibly too broad.

That also says where a difference would show. The miss is largest where the stretch changes fastest, toward the far shore, so a pond drawn by the scroll’s own projection would look broader along its far side than an ellipse and tighter along its near side. At a larger painted scale the same geometry scales with it: a drawing twice the size misses by twice as many pixels on a pond drawn twice as deep, and the proportion stays a tenth.

How far from an ellipse #

A pond that misses its best ellipse by three pixels is visibly not an ellipse only on close inspection, and it is worth knowing how the miss scales, because a garden’s ponds come in all sizes.

The miss grows faster than the pond. A pond half a metre across the radius departs from its best conic by 0.059 px, a pond of three metres by 3.08 px, a pond of four metres by 6.59 px, and the fitted slope on logarithmic axes is 2.25 — a little more than the square of the radius. The pinhole of the scroll’s columns draws every one of them as a conic to 2 × 10⁻¹¹ px.

The steepness has a reason, and it is the one the essay on hyperbolas gave for a line’s sag. A scroll’s departure from a pinhole comes from how much depth a shape spans relative to how far away it is. A small pond spans little depth, and across that little depth the division by depth is nearly linear, so the curve is nearly the conic a locally-linear map would give. A large pond spans a lot of depth, and the nonlinearity bends it. The departure is therefore a function of the pond’s radius over its distance, and it grows faster than linearly in that ratio.

The same argument predicts what distance does, and the next two drawings check it.

A smaller pond, and a further one #

Halving the radius to 1.5 m cuts the scroll’s conic miss from 3.08 px to 0.60 px — by a factor of about five, which is what a slope of 2.25 predicts. The pinhole’s widest-row error falls from 3.58 px to 0.86 px, by a factor of about four. The two errors belong to different drawings and obey different laws, and both vanish as the pond shrinks, which is why neither is noticed on a small pond drawn by either system.

Moving the three-metre pond from five metres to twelve cuts the scroll’s miss to 0.99 px and the pinhole’s widest-row error to 0.89 px. A pond at a distance of many times its own radius is drawn by both systems as nearly an ellipse with nearly the right widest row, and the differences this essay measures are confined to near, large ponds — a garden’s foreground rather than its far shore.

What a painter was given and what a painter did #

It is worth being careful at this point, because the numbers invite a claim they do not support.

The geometry says what a scroll’s projection does to a circle on the ground: a quartic, widest on its centre’s row, never leaning. It says nothing about how painters of handscrolls actually drew ponds. A painter drawing a pond in a scroll was not running a pushbroom; the pushbroom is a model of what a scroll’s accumulated viewpoint amounts to, established in the first essay on scrolls, and a painted pond is a painter’s decision about a shape. Whether painted ponds in scrolls are drawn widest at their middle, or as ellipses, or neither, is a question about particular paintings that no part of this essay measures.

What the geometry does establish is narrower and still useful. A painter who drew a round pond as a symmetric oval, widest across its middle, was drawing something much closer to what the scroll’s own projection gives than to what a pinhole gives — the widest row in the right place, no lean. A painter who drew it as a true ellipse was drawing the pinhole’s answer. If the drawn shapes are ever measured, the two predictions differ by a few pixels on a near, large pond, and that difference is the one worth looking for.

Still open: whether a scroll can range depth at all #

Everything measured about scrolls so far confirms what a map along, and a picture across concluded about depth in a scroll: it is absent. A column is one vertical line, and nothing relates one column to another in depth. One arrangement makes that stop being true without giving up anything a scroll keeps. Draw the same scroll twice, through two slits that lean a few degrees forward and back along the track rather than straight across it, and each point is drawn twice, from two places on the track. A scroll through two slits ranges in a straight line measures the separation between the two drawings of each point and finds it proportional to the point’s depth — not reciprocal to it, as every pinhole stereo pair is — with a depth error that is the same at every distance, and a cost paid instead in the length of the roll.