Diminution — where it appears
Named by 18 essays across 8 fields — each of them below, with the objects they name alongside it.
Dividing depth by eye
Three methods for spacing a receding row, all of them taught, all of them wrong. Read back as distances, the best misplaces a post by 3.5 metres in a row that is supposed to be spaced 1.4. And the obvious way of checking them gives one of the three a perfect score.
Stepping closer is not zooming
Changing the focal length leaves the ratio between any two things in a picture exactly alone — to twelve decimal places, at every focal length there is. Moving changes it. Hold the subject's drawn size across a step from 3 m to 1.5 m and the background halves, which is the whole of the shot everybody knows and nobody derives.
A picture with no size–distance signal
In a system with no diminution the drawn size of an object falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is. Depth has to be carried by something else, and what carries it is height on the page — linearly, and without a horizon.
A centre and a measure are exclusive
Eight drawing systems, measured on five questions, with the pinhole as a row rather than the header. Exactly one has a centre of projection and it is exactly the one with no true measure — and loosening the measure test by a hair lets it in, which is what says the boundary is real.
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 wall does not get darker as it goes away
The inverse square law is about a point source. A surface is not a point source, and the picture of a wall is exactly as bright at twenty metres as at two — the patch one pixel covers grows as the square of the distance and the light per unit area falls as the square of the distance, and a picture records the product. Which is why aerial perspective has to be the air.
Two grounds, and what the second one costs
The miniature convention wants its floor drawn from overhead and its figures drawn from in front, and the two optical axes it asks for are exactly ninety degrees apart. Read as a picture with two centres rather than as a picture with none, the arrangement stops being a contradiction and becomes a quantity: the rays of the composite miss their own best point by more than a metre, and the absorbed reading is a floor that leans.
A parallel floor under a perspective room
Draw the floor without diminution and the people on it with it, and the picture has a centre at infinity glued to a centre in the room. The same map absorbs it — but there is no shear this time, and there cannot be: bringing a point in from infinity is not something an affine map does, so the room the picture is equally a picture of has its midpoints moved as well as its angles.
A set cut for one eye
Build a colonnade four metres deep and cut every column so that its picture is the picture of one eighteen metres deep. The taper is forced rather than chosen — height and width both scale as the real depth over the intended one — and the match from the design eye is exact to sixteen decimal places. What gives it away is the second eye, and not by the ratio anybody would predict.
Nothing moves when the object does
Slide a box 3.2 m across the world and its parallel drawing is the same drawing translated 131.5 px — every edge the same length to 4e-14 px. The perspective drawing of the same move changes its edge lengths by 87.2%. One family's pictures depend on where a thing is; the other's do not, and almost everything the two families disagree about follows from that.
Size that means rank
In a great many pictures the drawn height records importance rather than distance. That is a decision rather than a mistake, and it can be caught with a straightedge by carrying one figure's height across the room by the taught construction and see where it lands. A tenth of rank in the picture already misses the drawn head by 21 px, the miss is exactly linear in how much rank is there, and the whole test needs two references and no arithmetic.
A picture in bands
A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.
A dolly zoom is a step and a zoom, and they meet at one depth
Step 1.5 m toward a subject 5 m away while shortening the lens to hold its size. Every mark moves along the line from the centre of the picture, to a ten-trillionth of a degree — outward if nearer than the subject, inward toward a limit if further, and not at all on the subject's own plane. The step and the zoom each move everything one way; the dolly zoom is where they cancel.
A dolly zoom off the axis keeps a line, not a plane
Step toward a subject along a track that is not quite the line of sight, and zoom to hold its size, and the plane that stood still in the classic shot stops standing still. The step now spreads from a point beside the centre while the zoom still shrinks toward the centre, and the two cancel only along one row of the picture, one depth per column. The subject itself slides by f·d·sin ψ over its distance — a pixel once the track is a quarter of a degree out — and turning to follow it holds the subject at the price of bending the rest of its plane, while shifting the frame instead holds the whole plane exactly.
Higher on the page, and where that stops being true
A pinhole's image height above the horizon falls monotonically with depth over the whole 2–400 m range sampled here, and 57% of that whole range lands in the ground's last drawn tenth — the accumulation the oldest depth convention is quietly built from. Above eye level the ordering inverts, and at eye level exactly, five different depths draw one height, a spread of 0.0e+0 px.
Two stations in one picture
A parallel floor under a perspective room found one map absorbing two centres into one sheet. Split a two-rule picture down the middle instead and each half hands back its own horizon — 7.80 px apart at a rule-mix of 0.02 — and no eye's position has anything to do with the gap, because an ordinary pinhole picture's recovered horizon does not depend on where the eye stood at all.
The map a row of posts is
Walking one bay further down a row is a map of the drawn line to itself, and which map it is settles everything about how the spacings behave. It is parabolic — one fixed point, counted twice, and that point is the vanishing point — which is why the drawn posts crowd toward it and never arrive. Doubling a distance instead gives a hyperbolic map whose multiplier a straightedge can read.
A straightedge convicts the rule on five braccia
The constant-ratio rule for spacing a pavement's boards leaves one mark a correct construction does not: its tile corners bow off the diagonal. The worry was that a painter's hand would bury the bow in its own scatter. At a pixel of scatter it does not, on any pavement of five braccia or more. What limits the test is not the hand but the page: a pavement drawn to one width bows no more at twenty braccia than at eight, while the hand's wander keeps growing with every tile.
Named alongside it
The objects these essays reach for when they reach for this one.
Depth cueForeshorteningDemonstrationHorizoncentre of projectionStation pointDrawing systemfield of viewFree parameterParallel projectionVanishing pointDepth compression