Incidence — where it appears
Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.
The third point put where it looks right
Three-point perspective is taught as two vanishing points plus a third placed by judgement. The third is not free: two points and the centre of the picture fix it exactly. It survives being placed by eye because in the layout a book draws it belongs thousands of pixels off the paper, where 400 px of error costs less than a degree — and in a photograph taken looking up at a tower the same 400 px costs ten.
Measured down from the waterline
Whatever stands so far above the water, draw its reflection the same distance below. Through a vertical picture plane that is not an approximation — it is the reflection, to the arithmetic floor. Tilt the camera twenty-two degrees and it is eleven pixels out. Draw a gull, which touches the water nowhere, and guessing its waterline point wrong by two metres of depth costs fifty.
Carrying a height across the room
A known height at one place on the floor, and the same height wanted at another: two lines settle it, and they settle it exactly, at every camera and every pair of positions. What the recipes never mention is that one of those two lines has to be drawn to a point that is usually not on the paper — 3,300 canvas widths away in the case drawn here — and that the repair is not to extend it further.
A picture with nothing straight in it
Every construction on this site is handed the horizon, and a photograph of a crowd, a hillside or a curved façade has no straight edge to give it. What such a picture does have is repetition — and three things of one height put the horizon exactly where the camera has it, from the picture alone. Two things do not, and three standing abreast do not either, and both refusals are the reader's ordinary situation.
Seven is not a power of two
Halving a receding depth by diagonals is exact, and every book gives it. Halving reaches a half, a quarter, three eighths — and never a third, however many times it is spent, because no power of two is divisible by three. There is a construction that reaches every whole fraction, it costs three lines rather than a stack of quadrangles, and the extra ingredient is not a measurement.
The conic a circle becomes
A circle photographed is an ellipse, or a parabola, or a hyperbola, and which one is decided by a single incidence: whether the circle reaches the plane through the eye parallel to the picture. Not the lens, not the tilt, not how far away it is. The discriminant of the image agrees with that one test at every point of a sweep, and at the crossing it is zero to 1e-13.
The line every nosing is on
Nothing in a staircase points up the pitch. Every surface in it is level or vertical, and its picture has a vanishing point off the horizon all the same — belonging to the line the front edges of the treads lie on. That point is not free: it is collinear with the travel point and the vertical point, and the angle it makes says the rise-to-run the builder chose.
The polar with a straightedge
Two secants through a point cut a conic at four places; the complete quadrangle they make has two more diagonal points; the line through those is the polar. Not one length, angle or midpoint is used, so the whole construction survives the projection that made the picture — and three unrelated pairs of secants land on the same line to 4.3e-13, while moving the point moves it by fifteen orders of magnitude more.
What a straightedge reaches on a receding line
Three marks on a receding line fix a coordinate, and everything a straightedge builds from them is a ratio of whole numbers. Three complete quadrangles put 1655 marks on the unit interval with no gap above 0.0038, a third costs two quadrangles where repeated halving never reaches one at all, and the same net built on a line that does not recede arrives at the identical 30 coordinates.
A point and a line are one object
Every projective statement has a partner got by swapping the words point and line, and the partner is true whenever the original is. Run on this collection's own Desargues configuration, all thirty of its incidences survive the exchange to 5 × 10⁻¹⁶ — and three of the dual's ten points land at infinity, which is a fact about where the drawing sits on the page rather than about the theorem.
Desargues read the other way
The theorem's converse is not a second theorem. Exchange points and lines in the ten-point configuration and every one of its thirty incidences holds to 5 × 10⁻¹⁶, the axis becomes a point, the three axis points become lines through it — and what has been written down is the converse, read off the drawing rather than proved.
Pascal's line, and the theorem underneath Pappus
Six points of a conic, three meets of opposite sides, collinear to 1.3e-12 px. Flatten the conic towards a pair of lines and the residual never leaves the axis — 5.7e-14 px at the end, where the theorem has become Pappus's — so Pappus is the degenerate case of Pascal reached continuously rather than a separate theorem that resembles it.
Every projectivity is two perspectivities
A perspectivity is what one eye does between two lines, and two of them compose to any projectivity at all. The construction closes on a point nobody used to 1.4e-14 pixels, both of its free choices move the second centre 663 pixels across the picture, and the composite does not move at all.
The theorem that is obvious one dimension up
Desargues in the plane needs a proof and in space needs none — two triangles in different planes have their corresponding sides meeting on the line where the planes cross, and the meets land 1.0e-14 metres off it. The plane figure is a shadow of the spatial one, and five different solids cast the same photograph to 2.0e-12 pixels.
Named alongside it
The objects these essays reach for when they reach for this one.
Vanishing pointHorizonDualityStraightedge constructionCross-ratiopoint at infinityComplete quadrangleConditioningcentre of projectionConicDesarguesGround plane