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

Back projection — where it appears

Sending a mark on a picture back out along the ray that made it, which recovers a direction and never a distance. It is the first half of every measurement made from a picture, and the second half is always some other constraint — a plane, a second view, a known length — that turns the ray into a point.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

horizondepicted height ÷ side0.3278corner angles90.000° — forced by the methoddepicted side ratio0.7195lens this drawing implies60° acrossdrawn exactly as the method prescribesthe free step is where the far edges go

The cube that is a box

The two-point cube every book teaches has a step it supplies no construction for. Place the two far edges symmetrically and the drawing depicts a square plan for free; place them eight points apart — a difference invisible on the page — and it depicts a box 1.4 times shallower than it is wide.

wrong · Cubebyeye near edge 155 pxfar edge 196 pxrecovered: ratio 1.000000, corners 1e-14° from squarecorrect from 12 cm, at 160 mm wideplane tilted -45°

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.

choices · Divergence floor: tilt 36.7°footstool: tilt 65.1°table: tilt 54.2°book: tilt 78.2°assumed camera 520 pxspread 41.4°

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.

choices · Divergence 2.7:13.1:14.5:15.8:17.2:1in plan; ellipses at 8σworst 7.2:1

The answer is an ellipse

A mark read with a round error does not come back as a round region on the ground. The ray is grazing, so a pixel across the picture is worth a little and a pixel up it is worth a great deal, and the recovered point's uncertainty is an ellipse pointing at the camera — 5.07 to 1 at eight metres from a camera 1.62 m up, which is the depth over the height. Propagated and sampled agree to 0.5 per cent, and a ray that is not grazing gives a disc.

metrology · Uncertainty correct from 20 cm, at 160 mm wideresidual 2e-15 m · 1/sin up to 1.22

A shadow across a second object

A straight edge held in front of a lamp defines one plane, and the shadow's boundary is wherever that plane meets something. That turns a picture of a shadow into a measurement: a camera ray and a known plane meet in one point, and the object the shadow is falling on comes back out.

light · Curvedreceiver plan: the apex has movedfree along the kernelisometricrecovered to 1e-16

The drawing that gives the solid back

One parallel view of a general point determines nothing: two equations, three unknowns, and the kernel is free. What closes it is not a second view but the correspondence — knowing which drawn edge runs along which world axis — and with it the whole solid comes back out of one drawing, exactly.

parallel · Ruler two lamps: what one centre leaves, floor by floora flat floor138.33 pxa dished floor138.33 pxa dished floor, k = 0.02138.33 pxa dished floor, k = 0.06138.33 pxa dished floor, k = 0.12138.33 pxa ridged floor138.33 pxa ridged floor, k = 0.02138.33 pxa ridged floor, k = 0.06138.33 pxa ridged floor, k = 0.12138.33 pxa floor with a step138.33 pxa floor with a step, k = 0.02138.33 pxa floor with a step, k = 0.06138.33 pxa floor with a step, k = 0.12138.33 pxtwo lamps, four floors, four curvaturesspread over all thirteen: 2.8e-14 px

A floor cannot fake a second lamp

Cast the same two lamps onto four floors at four curvatures and ask how well one centre explains the drawing. Every one of the thirteen answers is 138.3277 pixels — the same to fifteen digits, because a floor decides where along a ray the shadow's tip landed, and a line through a point and another point that has slid along it is the same line.

light · Curvedreceiver horizonwhere the rays meet — the reversed imagecorrect from 19 cm, at 160 mm widethe lamp has no image · rays meet to 2e-13 px

A lamp behind the camera

A light behind the photographer has no image — the projection refuses it — and the shadows it casts are in front of them, drawn as ordinary shadows. The construction that recovers a lamp from those shadows works anyway, meeting to a ten-thousandth of a pixel at the point the reversed divide puts it, and the taught reading of where the answer lies gets the case exactly backwards.

light · Lightrecovery

Named alongside it

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

reconstruction ambiguityDepicted rectanglePoint lightVanishing pointCamera tiltConditioningFocal lengthFree parameterGrazing incidenceHorizonInverse perspectiveInverse projection

All concepts