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

Plan view — where it appears

A drawing or reading that keeps only the two horizontal coordinates of a point, dropping its height. It is what makes a shadow on a curved floor mispredict, because the height it discards is exactly what the floor decided, and a reading in rays never forms one.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

the eye the picture is for1 faces no ray reaches9 steps, 18 planes, eye 1.65 m up17 used · 58.5% of the picture on risers

A stair does not use all its faces

A corner anamorph is two homologies and a flight of nine steps is eighteen, which is arithmetic and is the least of it. What a flight has that a corner does not is that which faces exist and which faces can be painted are different questions. From the top of a descending flight not one riser is reachable at any eye height, so half the planes are unpaintable by construction — and from the bottom of an ascending one, 58 per cent of the picture lands on risers that are 36 per cent of the surface.

viewing · Anamorph the lampthe occluder, and where the ray puts it backthe mark, on the floorwhere the plan puts it — 4 mm outa dished floor, k = 0.06by ray: exact · in plan: 4 mm

The floor is a choice of coordinates

Four rungs of this field have measured what a curved floor costs a shadow reading, in millimetres. It costs nothing. A shadow mark sits on the ray from the lamp through the occluder's edge, and where along that ray the floor caught it is a fact about the floor alone — so un-casting in rays returns the occluder to three parts in ten thousand of a millionth of a millimetre on a plane, a dish, a ridge and a step alike, while the same shadow read in the plan is wrong by up to 216 millimetres.

light · Shadowinverse 05010000.0500.1000.1500.200how strongly the floor disheshow far the recovered foot is from the lamp's (px)the foot, found in the planthe lamp, found in rays — exactthree posts on a dished floor, one drawing133 cm of lamp at k = 0.22

The lamp comes out in rays and not in plan

One drawing of three posts and their shadows yields two points, and a curved floor treats them completely differently. The lines through each post's top and its shadow's tip meet at the lamp's image to a ten-thousandth of a pixel at every curvature, because a top and a tip are two points of one real ray. The lines through each foot and the same tips meet 113 pixels from the lamp's foot — and the lamp placed from an exact point and a wrong one lands 1.3 metres away.

light · Lightrecovery stops dead — a real cornerturns at speed — the creasefour receivers, one wire, one lampplane 1+0 · dish 1+0 · ridge 1+0 · step 1+5

The corners a floor cannot add

A wire with no corner anywhere on it casts a shadow with one, wherever its tangent runs along the ray. That condition contains the lamp and the wire and no surface at all — so the same helix over a plane, a dish and a ridge draws shadows that differ by metres and each has exactly one corner. A floor with a crease draws five more, and the two kinds are separable by a hundredfold: a real corner is where the shadow stops dead, and a crease is where it turns at full speed.

light · Curveshadow 8 lamps · 20 mm counts as read8.5% of 9.0 m²

A floor is read along curves

Whatever a shadow says about the floor it landed on, it says only where the shadow is — and a shadow is a curve while a floor is a surface. Shadow curve length grows exactly linearly in the number of lamps, by a fitted exponent of 0.999, and the fraction of floor within two centimetres of one grows more slowly at 0.94, because the curves begin to overlap. At thirty-two lamps, seventy-one per cent of a nine square metre patch has still never had a shadow on it.

light · Twolamps 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 the picture plane, in planto the observer, 208 px further downcorrect from 12 cm, at 160 mm widestation 520 px

Two rules for one pavement

Vignola set out two rules for laying a tiled floor and asserted that they agree. Executed from the same ground line and the same free parameter they agree to 2e-13 px; executed from the numbers their own wordings invite they part by 24 px. The quantity that separates them is the distance the reader has to stand at, and neither rule names it.

construction · Tworules

Named alongside it

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

Receiving surfaceShadow projectionDevelopable surfaceReconstructionModel errorRay tracingResidualcentre of projectionConditioningHomologyHorizonLight recovery

All concepts