Pascal's theorem

Abstract: Introduction Projective geometry: Basic notions of projective geometry Conics Intersection of two conics Complex analysis: Riemann surfaces Elliptic functions The modular function Elliptic curves Poncelet and Cayley theorems: Poncelet's theorem Cayley's theorem Non-generic cases The real case of Poncelet's theorem Related topics: Billiards in an ellipse Double queues Supplement: Billiards and the Poncelet theorem Appendices: Factorization of homogeneous polynomials Degenerate conics of a conic pencil. Proof of Theorem 4.9 Lifting theorems Proof of Theorem 11.5 Billiards in an ellipse. Proof of Theorem 13.1 References.

Abstract: This paper investigates a classical problem in computer vision: Given corresponding points in multiple images, when is there a unique projective reconstruction of the 3D geometry of the scene points and the camera positions? A set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points. For two views, it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface. We give a classification of all possible critical configurations for any number of points in three images, and show that in most cases, the ambiguity extends to any number of cameras. The underlying framework for deriving the critical sets is projective geometry. Using a generalization of Pascal's Theorem, we prove that any number of cameras and scene points on an elliptic quartic form a critical set. Another important class of critical configurations consists of cameras and points on rational quartics. The theoretical results are accompanied by many examples and illustrations.

Abstract: We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.

Abstract: If on an oval in a projective plane a 4-point Pascal theorem, π, with fixed points U and V holds, then the oval is {(x,y) ¦xy=c} ∪ (O) ∪ (∞), with c ≠ O, in some Hall coordinatization. If for every 3 distinct points P, Q, R (not on UV; neither U nor V collinear with two of P, Q, R) there is through them a certain point set satisfying an extended version of π, then all these sets together with all lines not through U or V form the circles of a plane Minkowski (= pseudoeuclidean) geometry over a commutative field. π may be expressed in terms of Minkowski geometry. Together with incidence axioms derived from the protective incidence axioms, the Minkowski version of π characterizes the plane Minkowski geometry over a commutative field and is thus equivalent to Miquel's theorem.