Moving viewpoint — where it appears
Named by 28 essays across 3 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 frame is an interval
An exposure is not an instant, so a frame is an integral of projections and every moving point draws a streak. The streak is straight, because the image of a straight path is straight — and its length goes as one over the depth, so two objects at 3 m and 6 m blur by lengths in the ratio 2.000. No single kernel describes the frame.
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 landscape that changes its rule halfway up
A reader handed only the marks of a three-station landscape recovers each band's own camera height without being told any of them — 1.60 m, 4.00 m, 11.00 m, to 3.6e-15 m. What that same reader cannot recover across a join is a common ground: the next band's own marks read as a ground point 2.40 m away from the true one at the first seam, 7.00 m at the second.
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 turning frame can be straightened; a travelling one cannot
Read a frame row by row while the camera turns at a radian a second and every point is 21 px from where a global shutter would put it, at every depth alike. Turn each row's rays back and every point returns to six trillionths of a pixel, with no depth known. Travel at 3 m/s instead, and the best correction that needs no depth is exact at one distance and 21 px wrong at 2 m.
The stations are also a staircase
A level eye draws a ground point on a row that depends on the eye's height and the ground's only through their difference. So a landscape drawn from three stations at 1.6, 4 and 11 m over flat ground is, to 6 × 10⁻¹⁴ px across ninety-nine samples, the same picture as one eye at 11 m over ground stepped 9.4 m and 7.0 m above the far plain. The station heights a reader recovers are recovered on an assumption the marks neither supply nor test — and what would decide between the two readings is a cliff 103% of a band's own height, which is exactly where the convention lays its mist.
Turning and travelling blur different worlds
A subject 8 m away crosses the frame at 4 m/s, and the camera keeps it sharp over a thirtieth of a second. Turn to follow it and every still thing blurs by the same 13.5 px, whatever its depth. Travel beside it and the still world blurs as one over its depth — 49 px at 2 m, 1.5 px at 64 m — while everything moving with the subject is sharp at every depth.
A seam breaks direction, not size
An eight-metre road crossing the first join of a three-station landscape is drawn 215.4 px wide under either band's rule, identically, and a six-metre post 161.5 px tall under either — the eye's height cancels out of any size taken at one depth. What does not cancel is where those sizes sit. The road's edges are turned 23.2° from each other and the post's foot lands 64.6 px out of place, and a painter butting two bands can absorb the offset and can never absorb the turn.
A shadow decides which landscape it is
One sun over a several-station landscape is one sun: every band images the light's direction and its shadows' at the same page point, and each reports the altitude as 22.000000°. A shadow is still dislocated at the join, and the reading that could not be settled by any ground point is settled by one — because a ray crossing the seam has a riser to descend that the flat reading does not give it, and the two tips land 5.94 m and 11.7 px apart.
Where a seam is allowed to go
Putting a join further off cuts the offset it forces from 210 px to 24 and leaves the turn at 23.2° exactly, so emptiness is a painter's only defence. And emptiness is not counted in metres: one twelve-metre object blocks 6.3 per cent of a landscape's depth wherever it stands and between 1.0 and 35.6 per cent of the picture, so how much room a seam has is set by where the landscape is crowded rather than by how much.
A camera on a bend is sharp on a circle
A camera car rounding a 50 m bend at 10 m/s, aimed into the bend, blurs the still world everywhere except at the bend's centre — and under a pixel from 42 m to 63 m along its axis. Off the axis the sharp place comes nearer as the cosine of the bearing, on the circle through the camera and the centre. Above the ground only the vertical line through the centre stays sharp. Aimed along the road, the camera has no sharp distance at all.
A seam that bends trades a turn for a gap
A join laid along a river bank instead of across one depth was supposed to go round what crosses it instead of avoiding it. It can, at a price: the far band can be lifted by one amount only, so a seam that wanders in depth opens a gap between the bands — 3 px buys the first join 1.2 m of wander, and going all the way round a twelve-metre object costs 15.8 px. Across a hundred landscapes, bending opens under 0.4 per cent of the picture that a straight seam did not already have.
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 frame's shear knows travel only over depth
Read a frame row by row while the camera turns and travels, and every vertical post leans — the near ones more. The lean is the turn plus the travel over the post's depth, and that sum is all the frame holds: twice the travel past posts twice as far draws the same frame to eighteen decimal places. Two posts cannot separate turn from travel. A facade can, because a turn leans the edges of the frame more than its middle, but the two signals are 99.8 per cent alike, and reading them apart takes a pixel on every row.
A rolling frame on a bend is right at its centre
Read a camera car's frame row by row as it rounds a bend and every still point moves — except the bend's centre, at every height, and the horizon row, at every depth. Each mark's displacement is its streak scaled by its row time, so the rolling frame is undistorted wherever the global one is sharp. And the usual repair makes it worse: a gyroscope that undoes the turn row by row puts 0.73 px back into the centre and is a net loss everywhere nearer than twice the bend's radius.
A turn moves the picture, not the blind point
A vehicle that yaws for a moment between frames points its camera somewhere new without taking it anywhere new, and that is exactly what a turn cannot help with. The mark it is driving toward stays blind at every turn; the disc around it moves by two hundredths of a pixel for a 3° yaw; with a sway, a turn changes a mark's depth cost by 0.2 per cent. Aiming the camera off the road moves the blind point across the picture — and off it past 30° — but the thing it blinds is still the thing the vehicle is heading for.
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.
Steering swings a camera too little to see ahead
A vehicle yaws about its rear axle, so a camera mounted ahead of the axle is swung sideways whenever the vehicle steers — a real baseline, where a turn alone gave the blind point ahead nothing. Half a degree of yaw swings a windscreen camera 1.3 centimetres and a bumper camera 3.1. Straight ahead at eight metres a pixel then costs 96 and 40 per cent of the depth. The cost falls as one over the arm times the yaw, and a usable 10 per cent needs 7.2 metre-degrees: two degrees of steering every frame on a bumper. Ordinary lane-keeping gives a fraction of that.
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.
A rolling frame on a bend is a coarse curvature gauge
Read row by row on a bend, a frame leans every vertical by an amount proportional to one over its range less one over the bend's radius, so posts at several ranges lie on a straight line that crosses zero at the radius — exactly, to a thousandth of a metre. Off the axis the zero moves to R·cos of the bearing, the circle on which a camera on a bend is sharp. As a gauge it is coarse: one frame's eight posts read a 50 m bend as 45 m (35–74) with tops placed to half a pixel, and sixteen frames as 51 m (47–55). The slower the sensor reads, the better it measures the road.
Named alongside it
The objects these essays reach for when they reach for this one.
PushbroomHandscrollStation pointDepth uncertaintyinstrument limitPiecewise mapcentre of projectionElevationSeamDepth compressionDisparityFree parameter