2912. On a Proof of Desargues' Theorem for a Pencil of Conics

1960 ◽  
Vol 44 (349) ◽  
pp. 213
Author(s):  
E. J. F. Primrose
Keyword(s):  
Pencil Of Conics ◽  
Desargues Theorem

The surfaces whose prime-sections contain a

1934 ◽  
Vol 30 (2) ◽  
pp. 170-177 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Hyperelliptic Curves ◽  
Ruled Surfaces ◽  
Minimum Order ◽  
Rational Surfaces ◽  
Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

scholarly journals On Desargues Theorem

1944 ◽  
Vol 34 ◽  
pp. 17-19
Author(s):  
J. H. M. Wedderburn
Keyword(s):  
Analytical Methods ◽  
Geometrical Proof ◽  
Analytical Proof ◽  
Desargues Theorem ◽  
Degenerate Cases

The usual proofs of Desargues Theorem employ either metrical or analytical methods of projection from a point outside the plane; and if it is attempted to translate the analytical proof by the von Stuadt-Reye methods, the result is very long and there is trouble with coincidences. It is the object of this note to give a short geometrical proof which, in addition to the usual axioms of incidence and extension, uses only the assumption that a projectivity which leaves three points on a line unchanged also leaves all points on it unchanged. Degenerate cases are excluded as having no interest.

scholarly journals Two-valenced association schemes and the Desargues theorem

2019 ◽  
Vol 9 (3) ◽  
pp. 481-493
Author(s):  
Mitsugu Hirasaka ◽  
Kijung Kim ◽  
Ilia Ponomarenko
Keyword(s):  
Noncommutative Geometry ◽  
Association Scheme ◽  
Technical Condition ◽  
Association Schemes ◽  
Sufficient Condition ◽  
Desargues Theorem

AbstractThe main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any $$\{1,p\}$$ { 1 , p } -scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

scholarly journals A quadratic transformation of the three-dimensional space of all conics through two points of a projective plane

2009 ◽  
Vol 42 (3) ◽  
Author(s):  
Giorgio Donati
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Quadratic Transformation ◽  
Pencil Of Conics ◽  
Dimensional Projective Space ◽  
Three Dimensional Space

AbstractUsing the Steiner’s method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projective mappings are induced by quadratic transformations of the three-dimensional projective space of all conics through two distinct points of a projective plane.

Moulton Planes

1961 ◽  
Vol 13 ◽  
pp. 427-436 ◽  
Author(s):  
William A. Pierce
Keyword(s):  
Translation Plane ◽  
Affine Plane ◽  
Negative Slope ◽  
Arbitrary Field ◽  
Desarguesian Plane ◽  
Real Numbers ◽  
Classical Plane ◽  
Real Affine ◽  
Desargues Theorem ◽  
Ordered Pairs

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

Desargues' theorem in n-space

1960 ◽  
Vol 1 (3) ◽  
pp. 311-318 ◽  
Author(s):  
Sahib Ram Mandan
Keyword(s):  
Projective Space ◽  
Algebraic Approach ◽  
The Other ◽  
Desargues Theorem

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

scholarly journals A case study in formalizing projective geometry in Coq: Desargues theorem

2012 ◽  
Vol 45 (8) ◽  
pp. 406-424 ◽  
Author(s):  
Nicolas Magaud ◽  
Julien Narboux ◽  
Pascal Schreck
Keyword(s):  
Projective Geometry ◽  
Desargues Theorem
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