Plates on what survives being seen from a point, each one computed in the browser as you watch: a cone cut by a tilting plane into a circle, an ellipse, a parabola and a hyperbola, two spheres wedged inside it that touch the plane at the foci, and a lamp that throws all four curves as shadows of one ring; and a tiled floor whose parallel joints meet on the horizon, four points whose cross-ratio no projection changes, and two triangles seen from one point whose sides meet in pairs on one line, because in space they lie in two planes, and two planes meet in a line; and six points on a conic, whose hexagon has its opposite sides meeting on one line on an ellipse, on a hyperbola and on a pair of lines, and whose six tangents make a hexagon with its diagonals meeting in one point; and a polygon between two circles, corners on one and sides touching the other, that closes from every starting point or from none, between two ellipses as well; and the projective plane itself, the lines through a point, where any two lines meet once, a Möbius strip hides, and Boy's surface builds the whole in space with V − E + F = 1. Every plate keeps in its corner a measurement taken on the figure on screen: the cross-ratios and incidences that projection leaves unchanged, and the eccentricities and focal distances that agree with their formulae.
Plates
I–V
Loop
60 s each
Resolution
Matches the stage, up to 4.4 MP
Frames prerecorded
None
Film
In production
Plate I · 4 movements · 1:00 loop
Conic sections
Menaechmus found these curves around 350 BC by cutting cones, and Apollonius gave them their names. In 1822 Germinal Pierre Dandelin proved their focal properties with two spheres, each touching the cone along a circle and the cutting plane at a single point, using only the fact that tangents to a sphere from one point are equally long. Seen projectively, a cone is the bundle of lines through its apex, and every section of it is a view of every other. In the corner the eccentricity is read from five points of the curve on screen, and the distances are measured on the figure.
Plates
Notes
What projection keeps
Seen from a point, lengths, angles and ratios of lengths all change, and circles turn into ellipses, parabolas and hyperbolas. What stays is incidence, which points lie on which lines, and one number, the cross-ratio of four points on a line. The corner of the second plate measures those on the figure on screen as it moves; the corner of the first measures what does change with the view, the eccentricity and the focal distances, against the formulae that predict them.
Points at infinity
Desargues gave every family of parallel lines one extra point, where they all meet, and the points added to a plane this way lie on one extra line. With them, any two lines meet in exactly one point, a parabola is a conic that touches the line at infinity, and a hyperbola is one that crosses it.
Measured on screen
Eccentricities are read from a conic fitted through five points of the curve being drawn, vanishing points are found by intersecting the drawn lines, and the lengths in the corner are measured between the points as they are drawn. None of them is written in; they come out of the figure.