Related papers: Degenerations of Pascal Lines

Given six points $A,B,C,D,E,F$ on a nonsingular conic in the complex projective plane, Pascal's theorem says that the three intersection points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called a pascal,…

Let ${\mathcal K}$ denote a smooth conic in the complex projective plane. Pascal's theorem says that, given six points $A,B,C,D,E,F$ on ${\mathcal K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. This…

Given a sextuple of distinct points $A, B, C, D, E, F$ on a conic, arranged into an array $\left[\begin{array}{ccc} A & B & C F & E & D \end{array}\right]$, Pascal's theorem says that the points $AE \cap BF, BD \cap CE, AD \cap CF$ are…

Given six points on a conic, Pascal's theorem gives rise to a well-known configuration called the \emph{hexagrammum mysticum}. It consists of, amongst other things, twenty Steiner points and twenty Cayley-Salmon lines. It is a classical…

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…

Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$,…

The Pascal Multimysticum is a system of points and lines constructed with a straight edge starting from six points on a conic. We show that the system contains 150 infinite ranges (and 150 infinite pencils) whose projective coordinates are…

This paper is a study of the so-called `ricochet configuration' (or $R$-configuration) which arises in the context of Pascal's theorem. We give a geometric proof of the fact that a specific pair of Pascal lines is coincident for a sextuple…

A set $L$ of straight lines and a set $P$ of points in the Euclidean plane define an arrangement $\mathcal{A}$ = ($L$, $P$) of construction lines and registration marks, if and only if: (1) any point in $P$ is a point of intersection of at…

We prove general results which include classical facts about 60 Pascal's lines as special cases. Along similar lines we establish analogous results about configurations of 2520 conics arising from Mystic Octagon. We offer a more…

In this paper we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the Mystic Hexagon. We give explicit equations describing the conditions for $d+4$ points to lie on rational…

Suppose there are $n$ harmonic pencils of lines given in the plane. We are interested in the question whether certain triples of these lines are concurrent or if triples of intersection points of these lines are collinear, provided that we…

A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb P^3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case…

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show…

For a finite set $U$ of directions in the Euclidean plane, a convex non-degenerate polygon $P$ is called a $U$-polygon if every line parallel to a direction of $U$ that meets a vertex of $P$ also meets another vertex of $P$. We characterize…

We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type $\bold E_7$ singular point and compute…

We first review some topics in the classical computational geometry of lines, in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of the set of lines in R^3 interacting with $n$ objects of fixed description…

Let ${\mathcal A} = \{A_{1},\dots,A_{r}\}$ be a collection of linear operators on ${\mathbb R}^m$. The degeneracy locus of ${\mathcal A}$ is defined as the set of points $x \in {\mathbb P}^{m-1}$ for which rank$([A_1 x \ \dots \ A_{r} x])…

The topology of the intersection of three quadrics in Euclidean 6-space is studied using Kollar results. This needs an existence of a line without real points in the complex projectivisation of quadrics. We establish the existence of such a…

G\"unter Ziegler has shown in 1989 that some homological invariants associated with the free resolutions of Jacobian ideals of line arrangements are not determined by combinatorics. His classical example involves hexagons inscribed in…