秩序之下 · No.14 · 实时图版

Algebraic

代数曲线与曲面

Plates on the curves and surfaces cut out by polynomial equations, each one solved in the browser as you watch: Bézout's theorem, that a curve of degree m and a curve of degree n meet in exactly mn points once the points of tangency, the complex points and the points at infinity are counted, with a line sliding off an ellipse, an ellipse turning inside another, two circles and the two points at infinity that every circle passes through, and two cubic curves crossing in nine points; the ninth point, that every cubic through eight of the nine points where two cubics meet passes through the ninth, with the pencil of cubics through them, Pascal's theorem and the addition of points on a cubic curve, which comes out the same in either order; and ovals and bitangents, a quartic perturbed from two ellipses into four ovals, the most it can have, with its twenty-eight bitangents all real, then shrunk until it has no real points at all while four real bitangents remain; and the twenty-seven lines on a cubic surface, all of them real on Clebsch's diagonal surface, each meeting ten others, three through each of ten points, and paired into thirty-six double-sixes; and nodes, the points where a surface pinches into a double cone, as many as a surface of its degree can have, four on Cayley's cubic, sixteen on Kummer's quartics and sixty-five on Barth's sextic. The curves and the surfaces are drawn by a shader from their equations, pixel by pixel, and every point and line the theorems speak of is solved for again in every frame, in complex numbers; the corner counts what the theorem says must be there, including what cannot be seen.