The thread: Projected, not constructed — page 6
Three conics are one conic and a choice of horizon
Ellipse, parabola and hyperbola are one curve and three answers to which line is infinitely far away. Pitching one camera over one 6 metre circle walks through all three, and the crossing sits at 30.465545° by two instruments with different units — but the line that decides is not the horizon, and the popular name for the choice names the wrong one of the two lines a picture of a plane has.
Constructing a viewThe hook is the centre, and the eye is not
Dürer's string frame projects from a ring of iron driven into a wall, so its centre of projection has a position anybody can measure with a tape. A hook 120 centimetres behind a frame 56 centimetres wide makes a drawing correct from 34.3 centimetres shown 160 millimetres across, with its principal point 161 px off the middle of the sheet. The gridded veil projects from a head instead, and 30 millimetres of head moves its marks by 13.27 millimetres.
What survivesPascal'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.
What survivesA curved surface made of straight lines
A hyperboloid of one sheet carries two families of exactly straight lines, so a photograph of a cooling tower is full of straight lines bounding nothing flat — 28 of them here, drawn to 2.3e-13 px of straightness against a parallel of the same surface that bows 120 pixels. Both families' directions satisfy one asymptotic equation, so their vanishing points lie on one conic in the picture and not on two, to 1.6e-12 px with no fitting anywhere.
What survivesA line is a closed curve
The point at infinity is an ordinary point, so a projective line is a circle — and the consequence is about order. Betweenness broke in 21.1 per cent of ten thousand random projectivities and separation in none of them, and the zero is a reading rather than a blind instrument because a fold of the same circle breaks it 3,522 times.
What survivesThree kinds of map on a row of posts
A projectivity of a line has two fixed points, one, or none, and every one of the three is a picture this collection already draws. The three orbits are told apart by where they go — one piles onto a fixed point, one crawls, and the third returns after six steps and is 1.9e-11 pixels from where it started.
What survivesEvery 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.
What survivesThe 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.
Drawn confidentlyA tapered part meets at its apex
The sides of a turned part that narrows by one degree meet 54 pixels from the vanishing point of its axis, at the image of its apex, and a quarter of a degree already moves them 15. Drawn toward the vanishing point instead, a two-degree part loses nine tenths of its own taper. Flare it the other way by 5.9 degrees and a correct photograph prints its sides parallel; flare it further and they spread with depth.