Plan view — where it appears
Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
A walk's order is on the floor, not on the page
A narrative panel whose copies are moments of one walk holds its order after all — in where the copies stand, because a walker moves a step between moments. Joined by the shortest route on the page, six copies seen from a standing eye come back in the order walked 68 times in 100. Undo the floor first, with the horizon the copies themselves supply and the angle of view guessed anywhere from half to twice the truth, and the same rule returns it every time the walk allows — leaving one bit, the direction, which a figure facing its way supplies.
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.
Each copy's facing reads the walk past the place it doubles back
A walk that doubles back defeats every reading of a narrative panel from where its copies stand, because a later stop lands nearer an earlier one than a step. The way each copy faces is a short arrow on the floor, and a route that has to agree with the arrows as well as the distances follows the walk past the turn: on walks turning 70° between stops, facings read to 15° return the order 94 times in 100 where positions alone reach 59 — and say which end came first. But only once the arrows are undone onto the floor with the horizon the copies supply. Read straight off the page, an arrow pointing away from a standing eye is drawn nearly flat, and the same facings return 82.
A walker who stops is ordered by the way it turns
Three copies of a walker who pauses on one spot stand a few centimetres apart, and no reading of where they stand can order them. The way they face can — but not by the rule that followed a walk that keeps moving, which reads a three-copy pause turning 90° only a quarter of the time. A reader that ignores the direction of legs shorter than a step, and charges those legs instead for the turn between the copies' facings, reads it 92 times in 100 and costs a moving walk almost nothing. The middle copy of the pause is the hard one, and it is held only where the walker turned well clear of the error in reading each facing.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Receiving surfaceHorizonShadow projectionDevelopable surfaceIdentifiabilityPicture planeReconstructionDrawing conventionForeshorteningGround planeGround plane rectificationModel error