Handscroll — where it appears
Named by 15 essays across 2 fields — each of them below, with the objects they name alongside it.
A scroll is a camera that moves
A Chinese handscroll is not a picture with a wandering viewpoint or a picture with no viewpoint. It is the image of an eye that travels along a track and records one vertical line at a time, and that object has an exact geometry — orthographic along the roll, perspective across it.
A straight line in a scroll is a hyperbola
Under a pushbroom the image of a straight world line is a Möbius function of the paper coordinate, which is a rectangular hyperbola. It is straight exactly when the line holds its depth — so a curve in a handscroll is a depth signal rather than a stylistic one, and the sag is computable in pixels.
The centre a scroll does not have
Fit a common point to the rays of one section of a handscroll and it misses by metres. The miss is not a residual to be tightened — it is exactly the standard deviation of the eye's own track, it grows linearly with how much is unrolled, and it goes to zero only for a section of no width.
Every row is a different camera
A shutter that reads its rows one after another images each of them from wherever the camera was at that instant, so a frame is a stack of projections indexed by height — a handscroll with the roll running down the picture. Its rays miss their own best centre by the spread of the eye's track, at a ratio of 0.988, and a global shutter's meet to 2 × 10⁻¹⁶ m.
A map along, and a picture across
The midpoint of a segment lying along a scroll's length images to the midpoint of its image, exactly. The midpoint of a receding one lands 22% of the way off. One projection, two answers — and the direction that keeps measure keeps something a perspective picture never offers.
A scroll is not a panorama
Both draw straight world lines as curves, and one of them is a projection. A rotating eye keeps its centre exactly however far it turns; a translating eye has none at all. Curvature and centrelessness are independent properties, and conflating them is the standard mistake about both objects.
Three distances in one landscape
A landscape assembled from a low station for the near ground, a level one for the middle and a high one for the far gives its furthest band 6.9 times the picture one camera would allow it — because the image of a fixed depth interval falls as one over depth squared, and its own station gives that back. What it costs is a jump in the rate at which depth runs, 2.50 at the first join and 2.75 at the second, and three views from one height leave no seam at all.
A pond in a scroll is not an ellipse
A pinhole draws a round pond as an exact ellipse whose widest row is 3.58 px off the row of the pond's centre — the drawn-circle error every perspective textbook warns about. A handscroll draws the same pond widest exactly on its centre's row, and draws it as a quartic that no conic fits: the best ellipse misses it by 3.08 px. Each keeps what the other loses, and off to one side the pinhole's pond leans 9.67 px while the scroll's does not lean at all.
A scroll through two slits ranges in a straight line
Draw a scroll twice, through a slit leaning 10° forward along the track and one leaning 10° back, and every point appears in both drawings on the same row, separated by 9.169 px for every metre of its depth — at four metres and at fifty-two. Depth is proportional to that separation rather than reciprocal to it, so a pixel of error costs 10.9 cm at every distance, averaging leaves no bias, and there is no range past which the depth runs off to infinity. The price is paid in roll: a 100 m scroll ranges nothing past 283.6 m.
A scroll round a bend loses its straight-line depth
Draw a scroll through two slits leaning ±10° from a track that bends, and the separation that was 9.169 px for every metre of depth stops being proportional. Outside a 100 m bend it is 653.8 px at 256 m where a straight track gives 2347, and it never passes 907.6 px however deep the point; inside a 200 m bend it runs nearly three times ahead of depth and no slit reaches past 165.3 m. The two drawings still share their rows, and the scale along the roll becomes a function of depth.
A scroll can be asked its own radius
The two marks a bend leaves separate exactly. The along-roll scale alone fixes the angle in the disparity, so one point and a neighbour at its depth give back the radius and the depth in closed form — 200 m and 40 m returned to a part in 10⁹, with no search. The two answers are not equally held: a scale read one per cent too large under-reads the depth by one per cent and over-reads the radius by tan(φ − α)/α, which is 50 for a point ten metres from a five-hundred-metre bend. And a painter who evens the scale out by eye reports a gentler bend, never a bend that was never there.
A scroll of a climbing road measures its grade
Every reading of the two-slit scroll has leaned on its two drawings of a point sharing a row, because the eye is at one height at both moments. On a road that climbs they do not — and what parts them is the height climbed between the two moments over the reach, which on a straight climb is 2·f·g·sin φ for every point at every depth and height. The scroll does not lose its rows to a hill. It gains a third mark, a gradient meter that a level bend cannot counterfeit.
An eye that pitches with the road keeps its rows
An upright eye climbing a road parts each point's two drawings by the same few rows, and that offset reads the grade. Fix the eye to the vehicle instead, so it pitches with the road, and the offset vanishes exactly — for every point, at every depth and height. The grade has not gone. It has moved into the posts, which now lean by an amount that grows with their depth, and into one drawing, which can now read the grade on its own.
A band lifted by columns bends its lines
A curved seam in a several-station landscape opens a gap because the far band is one picture lifted by one amount. Lift it column by column instead and the seam closes exactly — and the gap goes into the band's lines. Shift each column and every road bends by twice the old gap, horizon and all. Draw each column from its own eye and a road bends by 8.1 px near the seam and 3.0 at the band's far edge, dying only as one over its depth; the horizon stays straight; and the next join inherits 37 per cent of the gap the first one lost.
A vehicle's pitch lags the road by its wheelbase
A camera fixed to a vehicle does not pitch with the road under it; it pitches with the chord between its wheels. Over a step from level to six per cent, a two-slit scroll's rows — silent on any steady grade — depart by up to 2.18 rows over 6.5 metres of road for a 2.7-metre wheelbase, against 4.26 over 4.0 for a camera that pitched at the point. The excursion's width is the eye's own chord plus about the wheelbase, and one line of posts reads the wheelbase back to ±8 centimetres. A vertical curve does not silence the rows either; it shrinks them as one over its length.
Named alongside it
The objects these essays reach for when they reach for this one.
Moving viewpointPushbroomDepth uncertaintyDisparityinstrument limitStation pointBaselinecentre of projectionDemonstrationForeshorteningParallel projectionPicture surface