Search

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

The geometrical peculiarity of the line of mutual intersection of second-order surfaces and tetrahedron is considered, which is the essence of Chasles’ theorem. 3D models have been developed, which vividly illustrate different variants of the theorem. The method of building 3D models is given, which allows to use those in an educational course on the theoretical basics of geometrical modeling. A proof for one of the theorem variants is given. A conclusion has been made on the necessity to develop illustrative examples for all the variants of the theorem and its universal proof. The work has been performed using 3D computer modeling in AutoCAD package and SolidWorks.

We analyze an elementary theorem of Euclidean geometry, the Droz-Farny Line Theorem, from the point of view of the foundations of geometry. We start with an elementary synthetic proof which is based on simple properties of the group of motions. The proof reveals that the Droz-Farny Line Theorem is a special case of the Theorem of Goormatigh which is, in turn, a special case of the Counterpairing Theorem of Hessenberg. An axiomatic analysis in the sense of Hilbert [14] and Bachmann [2] leads to a study of different versions of the theorems (e.g., of a dual version or of an absolute version, which is valid in absolute geometry) and to a new axiom system for the associated very general plane absolute geometry (the geometry of pencils and lines). In the last section the role of the theorems in the foundations of geometry is discussed.

Projective invariants are not only important objects in mathematics especially in geometry, but also widely used in many practical applications such as in computer vision and object recognition. In this work, we show a projective invariant named as characteristic number, from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property, two high-dimensional generalizations of Pascal’s theorem are given, one establishing the connection of hypersurfaces of distinct degrees, and the other concerned with the intersections of a hypersurface and a simplex.

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A,B in S sharing a colour there is a third point C in S, of the other colour, collinear with A and B. Then all the points in S are collinear. We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A,B in S sharing a colour there is a third point C in S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly's complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a 3-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem., Comment: 23 pages, 3 figures

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.

In this paper we study plane conic geometry, particularly different representations of geometric constructions and relations in plane conic geometry, with Cayley and bracket algebras. We propose three powerful simplification techniques for bracket computation involving conic points, and an algorithm for rational Cayley factorization in conic geometry. The factorization algorithm is not a general one, but works for all the examples tried so far. We establish a series of elimination rules for various geometric constructions based on the idea of bracket-oriented representation and elimination, and an algorithm for optimal representation of the conclusion in theorem proving. These techniques can be used in any applications involving brackets and conics. In theorem proving, our algorithm based on these techniques can produce extremely short proofs for difficult theorems in conic geometry.

We discuss in the context of Euclidean and projective geometry a recently rediscovered theorem concerning the concurrency of the main diagonals of a hexagon, and provide a detailed proof of its hyperbolic version.

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.

An approach to discovering new theorems in geometry using numerical examples is discussed. Four classes of theorems related to Pascal conics have been discovered by this approach. Two of them are new.

For Laguerre planes Artzy [1] showed that a 4-point Pascal theorem on an oval leads to a configuration π, consisting of six points and five regular circles, which is equivalent Miquel's theorem. We represent some similar incidence assumptions which are again equivalent to π in each Laguerre plane. Besides, a uniform denotation to characterize different kinds of miquelian theorems in Benz planes is suggested.

Pythagoras’ Theorem is one of the most fascinating results in the History of Mathematics. Although there are indications that the result was already known before by the Babylonians, was with the Pythagorean School that there was a formal demonstration of this theorem. As Loomis (1972), in 1940 were known at least 340 different demonstrations of the Pythagoras’ Theorem, whose enunciation is as follows: “In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are over each cathetus”. This article discusses Pythagoras’ Theorem and some generalizations, and introduces a new generalization of this important theorem.

(1965). Various Proofs of Pascal's Theorem. Mathematics Magazine: Vol. 38, No. 1, pp. 22-30.