Vanishing line — where it appears
Named by 21 essays across 6 fields — each of them below, with the objects they name alongside it.
The horizon, and the fraction
The horizon crosses every upright at the point of it that stands at the camera's own eye height — always, whatever the picture plane is doing. It crosses at the same *fraction* of the drawn height only when the plane is vertical: tilt by 6° and the fractions spread by 0.08 percentage points, tilt by 4° and 0.06. One statement is an incidence and survives; the other is a ratio and does not.
The ramp has its own horizon
Every plane has a vanishing line, and a ramp's is not the ground's. Its uphill edges meet on a line above the horizon, and the angle at the eye between that meeting point and the same direction taken level is the gradient — 22.0000° recovered against 22° built, out of the picture alone, with no scale, no ruler and nothing known about the scene except that the ground is level.
A mirror that is not parallel to the wall
Carry the depth in front of the glass an equal depth behind it, square to the wall. That is exact for a mirror hung parallel to the wall and 1.26 metres — 107 pixels — out for one turned 20°. Two invariants survive the turn instead, and one of the two nearly did not survive being tested, because it had been written in a form that could not fail.
Along a line of constant depth the page is affine
Stepping a texture by a constant amount per pixel is wrong across a receding floor and exactly right along any line of it that stays at one depth — and on every plane those lines run parallel to its own vanishing line. Turn a 120 px span 1° away from that direction and it is 0.79 px out; roll the camera a hundredth of a degree and a floor drawn to 30 m is out by 0.69 px on its worst scanline.
Two mirrors are three cameras
A photograph with two mirrors in it holds three views of the scene from three places, at baselines of 2.90, 3.10 and 2.26 metres. Two of the three pairwise geometries are mirror pairs and are skew; the third is a rotation by twice the angle between the mirrors, and it is not new evidence — five numbers read off the print rebuild it to 9.4 × 10⁻¹⁴ pixels, where a general three-view arrangement needs eighteen.
A tilted span walks a staircase
A span along a banked floor's constant-depth direction is exact, and a renderer visits pixels rather than the span. Snapped to the grid, a 120 px span at a 20° bank costs 0.577 px where the same span along a page row costs 13.26 — twenty-three times better — and it never rises above 1.22 px at any bank. The price is bookkeeping: a band of twenty-four such spans draws 53 of its 1,368 pixels twice.
The conic a circle becomes
A circle photographed is an ellipse, or a parabola, or a hyperbola, and which one is decided by a single incidence: whether the circle reaches the plane through the eye parallel to the picture. Not the lens, not the tilt, not how far away it is. The discriminant of the image agrees with that one test at every point of a sweep, and at the crossing it is zero to 1e-13.
Two lines at infinity
A picture of a plane has two of them and they are not the same line. One is the horizon, where the plane's own infinity went; the other is where the picture's coordinates run out. The words ellipse and hyperbola are about the second, and every scrap of metric information is on the first — so a circle whose photograph is a hyperbola calibrates exactly as well as one whose photograph is an oval, to 7e-14 of a degree.
Five facts that close the same gap
The gloss that one length has to come from outside the photograph names a single option, and there are at least five — a length on the ground, the camera's own height, a repeated object of unstated size, a standing object of known height, and the focal length with the horizon. All five give an answer on the same picture with the same noise; their spreads run from 0.36 to 3.60 per cent, a factor of ten, and which one is available depends on the photograph rather than on the arithmetic.
A drawing has three horizons
The triangle three vanishing points form is not a scaffold. Each of its sides is the vanishing line of one of the three coordinate planes — the floor's horizon and two walls' — and its orthocentre is the centre of the picture. So the horizon a perspective drawing is built on is one side of a triangle, and it is special only because the ground is where things stand.
Where each closure enters the stratification
The five facts that turn a photograph's ratios into metres do not all do the same job. Read on four quantities through the map each supplies — a cross-ratio, a ratio along a line, an angle, a length — three of them return all four exactly and two return only the first three. Set the free scale ten per cent wrong and the whole ten per cent appears in the length and nothing appears in the other three, to thirteen digits.
A wrong model shows at its own level
Give a single-view measurement the wrong length and the error stays in the length. Give it the wrong model — a slope where it assumed level ground, a horizon five pixels out, a focal length off the barrel — and the error lands on the level that model belongs to and on every level above it, never below. That is a diagnostic. Two equal things at different depths catch a wrong plane; two equal things at right angles catch a wrong shape; marks read to 0.4 px see a slope of 0.38°. A wrong length, alone, nothing in one picture can see.
A plumb wall tells a slope from a horizon, faintly
A sloping ground and a misplaced horizon fail the ground's depth test alike. A wall standing plumb beside the ground separates them in kind: a slope leaves the wall's test at 1e-16, and a horizon moves it. But it moves it a tenth as far as it moves the ground's — 0.20 per cent for a horizon five pixels out, against 2.11 — so a pair of windows a metre wide cannot see the difference, and a whole facade six metres wide can tell a one-degree slope from its matching horizon 95 times in a hundred. A wall running straight away from the camera sees nothing at all.
A facade's corners see the horizon its widths could not
Read by its level widths, a six-metre facade saw a misplaced horizon only when it was 11.5 pixels out. Read by its two corners — the same wall's plumb edges — it sees 4.2, as well as the ground does, and four posts beside it bring that to 2.4. The corners' vanishing point is nearly six thousand pixels below the picture, but what the lines measure is its reciprocal, and in pixels of horizon their precision does not change from a camera pitched two degrees down to one pitched eight.
A leaning post is worth its lean, not its length
A facade's two plumb corners read a misplaced horizon at 4.2 pixels; a street's lamp posts and poles are more numerous and never quite plumb. Each lean is random from post to post, so it averages away with the count and puts no floor under the reading — but it is large beside a long line's reading error, so it sets the price of every post: four steel columns leaning a twentieth of a degree match the two corners, and it takes sixty-four timber poles at a quarter. Counted as the corners' equals, eight such poles make the corners' horizon 2.4 times worse; weighted for their lean, they leave it where it was.
The ladder of assumptions is a ladder of conditioning
Push the four corners of a board by one pixel and read three quantities through the one recovered map. A cross-ratio does not move at all — it is read in the picture and never went through the map. A ratio of parallel lengths moves by a tenth of a per cent at twenty degrees of obliquity and by 1.6 per cent at seventy-eight. An angle moves by sixteen thousandths of a degree and by nine tenths. The stratification ladder is usually taught as a hierarchy of what is assumed; it is also a hierarchy of what a pixel costs.
The horizon has a pole
Every other duality in this collection is a matrix somebody picked. A calibrated camera fixes one nobody picked, and under it the horizon of a plane and the vanishing point of that plane's normal are pole and polar — to 2.7 × 10⁻¹² pixels. Run backwards, those two marks give the focal length with no known length, no right angle and no square anywhere in the scene.
The stair that turns has a vanishing point that moves
A spiral stair's treads are one rectangle turned by a constant angle, so every front edge has its own vanishing point and the twelve lie on one horizon to 2.8e-14 pixels. Four consecutive of them read a cross-ratio of 1.353720, and the formula behind it holds no focal length, no principal point and no eye — so a photograph gives the builder's 12° turn back.
A measuring point for a ramp
Stepping true distances along a receding line needs a measuring point, and every printed rule puts it on the horizon. On a 1 in 6.0 ramp the ramp's own point lands every tread to 1.2e-13 pixels and the ground's puts the sixth one-metre tread at 2.57 metres instead of six. A halfway construction separates the two halves of the mistake, and the wrong radius costs 0.083 metres of the 3.43.
Figures on a street that slopes
Equal-height figures have their heads on one line, and the taught rule says the line is the horizon. On a street rising at 8.33 per cent the heads are still collinear to 2.8e-14 pixels and the line is 58.9 pixels above the horizon — the street plane's own vanishing line. The taught rule loses 1.46 m of a 1.62 m figure at the far figure, and runs out of figure altogether at 19.4 m.
Three conics are one conic and a choice of horizon
Ellipse, parabola and hyperbola are one curve and three answers to which line is infinitely far away. Pitching one camera over one 6 metre circle walks through all three, and the crossing sits at 30.465545° by two instruments with different units — but the line that decides is not the horizon, and the popular name for the choice names the wrong one of the two lines a picture of a plane has.
Named alongside it
The objects these essays reach for when they reach for this one.
Horizonsingle-view metrologyVanishing pointCross-ratioAffine structurePicture planepoint at infinityProjective stratificationMetric structureModel errorProjective mapConic