Camera tilt — where it appears
Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.
One, two and three point are one construction
The names count how many vanishing points sit at a finite place in the picture, and the count is a fact about how the object is turned and the camera is aimed. Nothing about the method changes between them, and a vanishing point does not appear — it arrives from infinity.
An inverse perspective is a leaning plane
Ask a divergent construction what solid it depicts and it answers: a rectangle, four right angles, near edge equal to far. What the splay encodes is not the shape but the plane's tilt — and a real square on a plane leaning toward the camera really does photograph with its far edge wider.
The horizon is at eye level — if the picture plane is vertical
The horizon cuts every standing figure at the same fraction of its height however far away it is, which is the most immediately usable fact in the subject. It holds when the camera is level, and a twelve-degree tilt is enough to spread the fractions by more than a percentage point.
A carpet and the people on it
A Persian miniature draws the ground as though from above and the figures standing on it as though from in front. The two views want optical axes exactly ninety degrees apart, and the plan view does not shorten a standing figure — it replaces its height with its distance from the point under the eye.
Turning the cameras inwards
A stereo pair made by rotating two cameras toward a common point puts the same world point at different heights in the two pictures — up to thirty pixels here, on a frame of four hundred. Two eyes level with each other see every point at the same height, so a pair with vertical difference is a pair of pictures of no scene at all.
A tile is an off-centre frustum
Rendering a picture in tiles is exact, and the way to do it is one line of arithmetic: a tile's sides are the whole frustum's sides read at the tile's own pixel bounds. Aiming the camera at each tile instead is defensible at every step and is a different picture, out by about a tenth of a tile whatever the tile size.
Four surfaces, and no one camera that draws them
Read under one assumed camera, the floor, footstool, table and book of a constructed divergent picture imply tilts of 36.7°, 65.1°, 54.2° and 78.2°, where one camera photographing four parallel surfaces gives each of them 35.0°. But the spread between the tilts is 46.8° under a 260 px lens and 6.7° under a 5,000 px one, so it measures the lens as much as the picture. The measure that owes nothing to a lens is on the page: the nearest drawing one camera could make moves the far corners by 17.6 px.
One camera means one horizon, not one point
The test this field has been using asks whether a picture's surfaces share a meeting point. One camera photographing four parallel surfaces turned by different angles in their own planes gives them meeting points 1,065 px apart in column and identical in height to 3 × 10⁻¹² px — so the shared-point test charges 28.7 px to a picture one camera really took, and the charge grows with the turn. What one camera imposes is a shared vanishing line. The earlier verdicts survive intact, and for a narrower reason than they looked to have.
The rows under a splay measure the bays, not the lean
A splayed strip drawn with its rows evenly spaced is still an exact picture of one flat plane leaning toward the eye — the straight sides fix the plane, whatever the rows do. What the rows decide is how deep each bay is, and evenly spaced ones make the near bay deeper than the far by nearly the square of the splay. That, and the six-per-cent gap between even rows and a camera's, depends on the splay alone — not on the lean, the distance or the lens.
The rows count hands, not cameras
Rows drawn between two straight sides charge a camera nothing — every strip of a four-strip divergent picture reads back as a flat plane, whatever placed them, so a single viewpoint redraws the whole picture exactly. What they do fix is one number per strip, and that number survives the lens. Read the same drawing at focal lengths twenty to one apart and the lean runs from 36.4° to 86.0° while the habit stays at 1.000000000.
A vanishing line with a slope in it
Turn the plane about the view direction and no family of any surface's edges is level; each vanishing line acquires a slope, and a group must agree about two numbers rather than one. The count does not change character — a hand of four pixels costs the test 3.00 px at no slope and 3.27 at thirty-eight degrees of it. What the slope does expose is the redraw: holding each far edge at its drawn height charges 0.95 px to a picture one camera really took.
Three-point, laid out with a straightedge
Recovering a camera from a drawing is the familiar direction. The other direction — stand somewhere, measure a room, and lay the picture out — had never been taken in three-point, because the third axis needs a measuring point on a line nobody draws. With it, every corner lands where the camera puts it to three parts in ten million million of a pixel, and nothing anywhere is judged.
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.
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.
The room a divergent picture is a photograph of
A divergent construction depicts a rectangle on a plane leaning toward the camera, and the splay alone sets how far — 36.3° at a splay of 1.32. Stand a second such construction on the first one's far edge, as a wall, and the two recovered planes meet at 6.9°, not at the right angle a real room's corner would need.
Named alongside it
The objects these essays reach for when they reach for this one.
Vanishing pointDepicted rectangleInverse perspectiveone-point perspectiveHorizonPicture planeDemonstrationFocal lengthPrincipal pointProjective invariantProjective mapreconstruction ambiguity