Visual Plane
every picture states the point it is correct from
Concept

Subtended angle — where it appears

The angle something occupies as seen from a point, which is what a picture records and a length is not. It is why a picture cannot supply a size, and why a reference length in the scene is the one thing that turns its ratios into measurements.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

full frame · 39.6°APS-C · 26.6°Micro Four Thirds · 19.6°1 inch · 15.0°phone (1/1.7″) · 8.7°one 50 mm lens · the angle is a property of the rectangle behind itMicro Four Thirds: 2.00× diagonal, 2.08× wide, 1.85× tall50 mm across five formats39.6° down to 8.7°

A focal length is not an angle

Fifty millimetres means nothing until a rectangle is named behind it. The same lens is 39.6° across full frame, 26.6° across APS-C and 8.7° across a phone sensor — and the distance the resulting print is correct from depends on the ratio of the two, so two cameras matched on angle agree exactly whatever their formats.

sensor · Format 50 mm on full frame · 39.6° acrossphone3.71×94 mm correctlaptop1.28×431 mm correct27-inch monitor0.78×829 mm correcttelevision1.52×1.7 m correctcinema0.84×16.7 m correct×1 — standing at the station pointhow many times further away the reader is than the picture's own station pointworst is the phone at 3.71×

The screen sets the distance

Every viewing distance quoted for a picture on a page is conditional on an assumed figure width. Replace the assumption with an actual chain — focal length, sensor width, display width — and the same 50 mm frame is correct from 9 cm on a phone, 83 cm on a monitor and 16.7 m in a cinema. Nobody is standing at any of them.

screen · Station 051050100150field of view the picture was rendered at — degreeshow many times the depicted depth is stretchedthe screen subtends 49.3°100° → depth ×2.6027-inch monitor at 650 mmsubtends 49.3°

A wide field on a small screen

A picture rendered at a hundred degrees and shown on a screen that subtends forty-nine is being read from two and a half times its own station distance, so the depicted space is two and a half times too deep. The stretch at the edges everybody complains about is correct; the complaint is really that nobody is sitting where it would be invisible.

screen · Screenfov the piazza, in planthe Baptistery, 25.6 m53 m27.2°the cathedral doorthe panel, and the eye it needsthe Baptistery fills 100%the eye is 60 cm back — off this sheet290 mm panel · 27.2° of Baptisterycorrect from 60 cm · 27° across

Brunelleschi drilled a hole in his panel

The first perspective demonstration in the European record came with its viewing point enforced — a hole through the back of the panel, a mirror held out in front, and one place to stand. That distance is computable from the panel's size and the angle the Baptistery subtends, and the answer lands squarely on the arrangement the account describes.

construction · Station 00.50011020304050distance from the camera to the wall (m)relative to the value at 1 mthe patch, growing as d²the light per unit area, falling as 1/d²their product — what the picture records2500× the footprint at the far endproduct flat to 2e-16

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.

light · Radiance 0204051015distance from the source to the ball, in radiifraction of the ball's surface that is lit (%)one half — the source at infinity25.0% at 2 radiicurve: the closed form · dots: quadratureagreeing to 5e-4

A lamp lights less than half a ball

Everyone knows a sphere is half lit. It is half lit by a source at infinity and by nothing else: a lamp two radii away lights a quarter of it, and the boundary it draws is a circle offset toward the light rather than a great circle. The offset is R²/D, which means a photograph of a lit ball carries the distance to whatever lit it.

light · Terminator -10100.5001the screen's radius, log₁₀ metresvertical difference between the eyes, log₁₀ arcminutesa stated fusion limit, 15′eyes 63 mm apart, at each screen's own sitting distanceunder the limit throughout

What the two eyes are sent

A reader's eyes are two seats sixty-three millimetres apart, so a curved screen delivers each of them a different map — and the part of the difference no homography absorbs is binocular evidence of the glass. Turned into a depth it comes back as the screen's own sag, 49 millimetres against 47 on a television, by a route that never saw the radius.

screen · Stereodisplay 11.201.401.6001234how far along the sofa the seat is (m)largest pixel over smallest, across the picturecurvedflat, same widthcurved television against a flat panel of the same widththe curved worst case is at 1.5 m, not at the end

The evenness a curve buys

A curved screen is sold on evenness, and evenness turns out to be three quantities that disagree. On pixel pitch the curve wins from every seat; on the angle the glass is turned through it wins until three and a half metres along the sofa; on the plain distance from eye to glass — the reading the argument is usually made in — it gives up before half a metre.

screen · Screenfov 89°in plan · the mark's size is the angle at the pointthe track covers 60°

The spread a point gets

A track of sixty degrees gives no point of the scene sixty degrees. The nearest receive 89 and the furthest 41, a factor of 2.2, and their errors run from 1.5 to 8.4 millimetres — following the angle at the point as its −1.68 power, with 94 per cent of the variation explained. The arc a track covers is one number for forty-four different situations and predicts none of them.

manyviews · Spread where the eyes are aimedin plan, framed to the shells drawnzero at 0.92 m, 40° aside

Vergence moves the shells and does not respace them

Turn two eyes inward and the depths a whole-pixel reading can report stop being planes and become a family of near-circles through both eyes — the twenty-pixel shell standing at 0.74 m forty degrees aside where a parallel pair puts it at 3.82. The spacing between consecutive shells is the same to 0.07 per cent across the whole field, so vergence relabels the rays and does not sharpen them, and the resolution argument for turning the eyes in does not exist.

depth · Disparity in plan · ellipses 25× actual sizewithin 8.4° of the mean sight

Split the track and the needles turn

A point triangulated from a track of cameras is not known equally well in every direction: its error is a needle, long along the direction its lines of sight barely constrain. That direction is not the mean line of sight but the lines of sight weighted by how near each camera is — and it can be turned. Two groups of cameras more than a right angle apart swing every needle across, and the same six pictures of a facade then measure its depth four times better.

manyviews · Spread

Named alongside it

The objects these essays reach for when they reach for this one.

Demonstrationfield of viewViewing distanceViewing positionDepth cueDepth uncertaintyStation pointBinocular disparitycamera trackConditioningCovarianceDepth compression

All concepts