Pascal's Theorem Research Articles

Degenerations of Pascal lines

Let $\mathcal{K}$ denote a nonsingular conic in the complex projective plane. Pascal's theorem says that, given six distinct 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. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal's hexagrammum mysticum.

A Pascal's theorem for rational normal curves

Pascal's theorem gives a synthetic geometric condition for six points a , … , f in P 2 to lie on a conic. Namely, that the intersection points a b ¯ ∩ d e ¯ , a f ¯ ∩ d c ¯ , e f ¯ ∩ b c ¯ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d + 4 points in P d to lie on a degree d rational normal curve? In this paper we find many of these conditions by writing in the Grassmann–Cayley algebra the defining equations of the parameter space of d + 4 -ordered points in P d that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.

A Hyperbolic Proof of Pascal’s Theorem

We provide a simple proof of Pascal's Theorem on cyclic hexagons, as well as a generalization by Mobius, using hyperbolic geometry.

Generalised Manin transformations and QRT maps

<p style='text-indent:20px;'>Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root.

Графические алгоритмы построения квадрики, заданной девятью точками

The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of any number of points of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J.H. Engel’s well-known algorithm, which makes it possible to define the ninepoint quadric metric. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem. The article discusses two well-known graphical algorithms for constructing a quadric (the Rohn — Papperitz algorithm and the J.H. Engel algorithm) and proposes a simplified version of the J.H. algorithm. For its constructive implementation using computer graphics. All algorithms allow you to determine the set of points and the set of flat sections of the surface of the second order, given by nine points. The Rohn — Papperitz algorithm, based on the spatial configuration of Desargues, is best suited for its implementation on an axonometric drawing using 3D computer graphics. Algorithm J.H. Engel allows you to solve a problem on the plane. The proposed simplified constructive version of the algorithm J.H. Engel is supplemented with an algorithm for constructing the principal axes and symmetry planes of a quadric, given by nine points. The construction cannot be performed with a compass and a ruler, since this task reduces to finding the intersection points of two second-order curves with one known general point (third degree task). For its constructive solution, a computer program is used that performs the drawing of a second order curve defined by an arbitrarily specified set of five points and tangents (both real and imaginary). The proposed graphic algorithm can be considered as an alternative to the algebraic solution of the problem.

Особенности свойств параболы при ее моделировании

When studying the theory of contour construction in “Affine and Projective Geometry” course on educational program specializations “Computer-Aided Design Systems” and “Applied Informatics in Design” a unit of computational and graphic task "Contour Construction" is carrying out in structural design. In this computational and graphic task the contour constructions are carrying out by second-order curves (a circle — by the radius and graphical method; a hyperbola, an ellipse, a parabola — by means of Pascal curves, taking into account positions of engineering discriminant). The constructions of an arc of ellipse, hyperbola, and parabola are carried out based on Pascal theorem: in any hexagon, which vertices belong to a second-order series, three points of the opposite sides’ intersection lie on one straight line — the Pascal line. However, in construction of a conic (a second-order curve), it is necessary to draw students’ attention to the fact that the points belonging to a second-order series (a second-order curve, or a conic) make a geometrical locus of intersection of Pascal hexagon’s adjacent opposite sides. By this method students successfully construct conjugate arcs of an ellipse and a hyperbola with other conics. The construction of a parabola arc, conjugated with other conics, is carried out by the method of engineering discriminant (it is more convenient to divide line segments in halves: a median and a triangle side, which is opposite to its vertex lying on a parabola arc). It should be noted that theoretical and practical material on this subject corresponds to the assimilation of Study Plan’s necessary competences (in accordance with each educational program), however, some aspects of this subject are accepted by students simply by trust. The aim of this paper is research of construction methods for parabola, applied to contour simulation.

On the Reconstruction Problem for Pascal Lines

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 collinear. The line containing them is called the Pascal of the array, and one gets altogether sixty such lines by permuting the points. In this paper we prove that the initial sextuple can be explicitly reconstructed from four specifically chosen Pascals. The reconstruction formulae are encoded by some transvectant identities which are proved using the graphical calculus for binary forms.

A Simple Proof of Poncelet's Theorem (on the Occasion of Its Bicentennial)

We present a proof of Poncelet's theorem in the real projective plane which relies only on Pascal's theorem.

From Pascal's Theorem to d-Constructible Curves

We prove a generalization of both Pascal's Theorem and its converse, the Braikenridge–Maclaurin Theorem: If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve C of degree d, then the remaining k(k − d) points lie on a unique curve S of degree k − d. If S is a curve of degree k − d produced in this manner using a curve C of degree d, we say that S is d-constructible. For fixed degree d, we show that almost every curve of high degree is not d-constructible. In contrast, almost all curves of degree 3 or less are d-constructible. The proof of this last result uses the group structure on an elliptic curve and is inspired by a construction due to Möbius. The exposition is embellished with several exercises designed to amuse the reader.

Obtaining a 3D extension of Pascal theorem for non-degenerated quadrics and its complete configuration with the aid of a computer algebra system

Pascal’s classic theorem states that: “the three intersection points of opposite sides of a closed hexagonal line, inscribed in a nondegerated conic, are collinear”. The following extension of Pascal theorem to 3D is considered: “given a closed decagonal line, inscribed in a nondegerated quadric, whose opposite side-lines are secant, the five intersection points of opposite side-lines are coplanary”. (A polygonal line with 10 sides is considered, because 10 — 1 points determine a quadric, as 6 — 1 points determine a conic). Obviously, in this extension to 3D of Pascal theorem some vertices of the polygonal line can not be freely chosen, but an interesting property has been found: the five diagonal lines passing through opposite vertices share a point. This property leads to a simple method to generate the configuration. Moreover, conditions of existence of this configuration are determined and the so called complete configuration is also described in detail. As large expressions appear when coordinates are used, we have developed a package on a computer algebra system that helps us to find and to automatically generate this configuration.

Curves in Cages: An Algebro-Geometric Zoo

The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of lines selected from two given finite families of cardinality d. The union of all lines from both families is called a cage, and the intersection X consists of d^2 points at which a line from the first family intersects a line from the second. The points of X are called the nodes of the cage. We study the subsets A of the nodal set X such that any plane algebraic curve C of degree d that contains A must contain X as well. As a corollary, we get a few generalizations of the famous Pascal theorem, generalizations that employ polygons (instead of hexagons as in Pascal's theorem) inscribed in a quadratic curve. Our results are closely related to the classical theorems of Chasles and Bacharach. Although the nodal sets X produced by our cages are quite special in comparison to more general complete intersections studied by Bacharach, the cages provide us with a much better grip on the combinatorics of subsets A of X with the property that is described above.

Some Incidence Theorems and Integrable Discrete Equations

Several incidence theorems of planar projective geometry are considered. It is demonstrated that generalizations of the Pascal theorem due to Mobius give rise to the double cross-ratio equation and the Hietarinta equation. The construction corresponding to the double cross-ratio equation is a reduction to a conic section of some planar configuration (203154). This configuration provides a correct definition of the multi-dimensional quadrilateral lattices on the plane.

Critical Configurations for Projective Reconstruction from Multiple Views

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.

Application of an incidence theorem for conics: Cauchy problem and integrability of the dCKP equation

We demonstrate that Pascal's classical theorem for hexagons inscribed in conics allows one to define in a compact manner maps which are governed algebraically by the integrable discrete CKP equation. A theorem for conics on oriented triangulated surfaces is used to construct a well-posed Cauchy problem for these dCKP maps. Moreover, the same theorem is exploited to construct in a purely geometric manner a Backlund transformation for dCKP maps. Thus, the integrability of dCKP maps and their underlying nonlinear soliton equation is shown to be encoded in an incidence theorem of projective geometry.

88.25 An algebraic proof of Pascal's theorem

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The Pascal theorem and some its generalizations

We present two generalizations of the famous Pascal theorem to the case of algebraic curves of degree 3.

A geometric approach for the analysis and computation of the intrinsic camera parameters

The authors of this paper adopted the projected characteristics of the absolute conic in terms of the Pascal's theorem to propose an entirely new camera calibration method based on purely geometric thoughts. The use of this theorem in the geometric algebra framework allows us to compute a projective invariant using the conics of only two images which expressed using brackets helps us to set enough equations to solve the calibration problem. The method requires restricted controlled camera movements. Our method is less sensitive to noise as the Kruppa's-equation-based methods. Experiments with simulated and real images confirm that the performance of the algorithm is reliable.

Pascal's Theorem and Quantum Deformation

By a transfer principle Pascal's Theorem is equivalent to a theorem about point pairs on the real line. It appears that Pascal's Theorem is equivalent to the vanishing of a common invariant of six quadratic forms. Using the q-deformed invariant theory of \cite{L} we construct corresponding quantum invariants by a computer calculation.

Tritangent centres, Pascal's theorem and Thebault's problem

Thebault's problem was posed in 1938, but a solution did not appear until 1983. Since then other solutions of the original problem and of various generalisations have been given. In this paper we discuss the generalisations (including some new results) and various background theorems on tritangent centres, making extensive use of Pascal's theorem.

Automated production of traditional proofs for theorems in Euclidean geometry I. The Hilbert intersection point theorems

We present a method which can produce traditional proofs for a class of constructive geometry statements in Euclidean geometry. The method is a mechanization of the traditional area method used by many geometers. The key idea of our method is to eliminate dependent (constructed) points in a geometry statement using a few basic geometry propositions about the area of triangles. The method has been implemented. Our program, calledEuclid, can produce traditional proofs of many hard geometry theorems such as Pappus' theorem, Pascal's theorem, Gauss point theorem, and the Pascal conic theorem. Currently, it has produced proofs of 110 nontrivial theoremsentirely automatically. The proofs produced byEuclid are elegant, short (often shorter than the proofs given by geometers) and understandable even to high school students. This method seems to be the first that can produce traditional proofs for hard geometry theorems automatically.