2012/01/06 by Zhongxuan Luo, Luo, Zhongxuan
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Numerical Analysis (math.NA) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1201.1344
openalex publication_date 2012/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1640's, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a different comprehension to Pascal's mystic hexagram or to the Pascal theorem. Using this invariant, the Pascal theorem can be generalized to the case of cubic (even to algebraic curves of higher degree), that is, \em For any given 9 intersections between a cubic Γ3 and any three lines a,b,c with no common zero, none of them is a component of Γ3, then the six points consisting of the three points determined by the Pascal mapping applied to any six points (no three points of which are collinear) among those 9 intersections as well as the remaining three points of those 9 intersections must lie on a conic. This generalization differs quite a bit and is much simpler than Chasles's theorem and Cayley-Bacharach theorems.