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Journal articles on the topic 'Desargues configurations'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

Chipalkatti, Jaydeep. "Pascal’s Hexagram and Desargues Configurations." Mathematica Pannonica 27_NS1, no. 1 (2021): 21–31. http://dx.doi.org/10.1556/314.2020.00004.

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This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to six conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

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2

Knarr, Norbert, Bernhild Stroppel, and Markus J. Stroppel. "Desargues configurations: minors and ambient automorphisms." Journal of Geometry 107, no. 2 (2016): 357–78. http://dx.doi.org/10.1007/s00022-015-0311-1.

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3

Berceanu, Barbu, and Saima Parveen. "Fundamental group of Desargues configuration spaces." Studia Scientiarum Mathematicarum Hungarica 49, no. 3 (2012): 348–65. http://dx.doi.org/10.1556/sscmath.49.2012.3.1210.

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We compute the fundamental group of various spaces of Desargues configurations in complex projective spaces: planar and non-planar configurations, with a fixed center and also with an arbitrary center.

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4

Dijksman, E. A. "Assembling Complete Pole Configurations for (Over)Constrained Planar Mechanisms." Journal of Mechanical Design 116, no. 1 (1994): 215–25. http://dx.doi.org/10.1115/1.2919350.

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The complete pole configuration of a planar n-link mechanism having one instantaneous degree of mobility, possesses (3n−4)/2 independent poles determining (n−2)2/2 remaining poles of the configuration. The dependency is demonstrated through Desargues’ Theorem and her generalizations. Simultaneously, pole configurations have been “elated” into three-dimensional point-lattices intersected by a plane. The insight obtained in these configurations allows the designer to find clues in building overconstrained linkage mechanisms meeting certain geometrical properties.

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5

Bruen, Aiden A., and James M. McQuillan. "Desargues configurations with four self-conjugate points." European Journal of Mathematics 4, no. 3 (2018): 837–44. http://dx.doi.org/10.1007/s40879-018-0274-5.

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6

Avritzer, D., and H. Lange. "Desargues Configurations via Polar Conics of Plane Cubics." Geometriae Dedicata 120, no. 1 (2006): 59–72. http://dx.doi.org/10.1007/s10711-005-9035-y.

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7

Ivashchenko, Andrey Viktorovich, and Tat’yana Mikhaylovna Kondrat’eva. "Projective configurations in projectivegeometrical drawings." Vestnik MGSU, no. 5 (May 2015): 141–47. http://dx.doi.org/10.22227/1997-0935.2015.5.141-147.

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The article focuses on the optimization of the earlier discussed computer method of obtaining new forms of polyhedra based on projective geometry drawings (trace Dia- grams).While working on getting new multifaceted forms by projective geometry methods based on the well-known models of polyhedra on the first stage of the work it is required to calculate the parameters of projective geometry drawings, and then to build them. This is an often used apparatus of analytical geometry. According to it, at first the parameters of the polyhedron (core system of planes) are calculated, then we obtain the equation of the plane of the face of the polyhedron, and finally we obtain the equations of lines - the next plane faces on the selected curve plane. At each stage of application such a method requires the use of the algorithms of floating point arithmetic, on the one hand, leads to some loss of accuracy of the results and, on the other hand, the large amount of com- puter time to perform these operations in comparison with integer arithmetic operations.The proposed method is based on the laws existing between the lines that make up the drawing - the known configurations of projective geometry (complete quadrilaterals, configuration of Desargues, Pappus et al.).The authors discussed in detail the analysis procedure of projective geometry draw- ing and the presence of full quadrilaterals, Desargues and Pappus configurations in it.Since the composition of these configurations is invariant with respect to projec- tive change of the original nucleus, knowing them, you can avoid the calculations when solving the equations for finding direct projective geometry drawing analytically, getting them on the basis of belonging to a particular configuration. So you can get a definite advantage in accuracy of the results, and in the cost of computer time. Finding these basic configurations significantly enriches the set of methods and the use of projective geometry drawings.

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8

Glynn, David G. "A note on Nk configurations and theorems in projective space." Bulletin of the Australian Mathematical Society 76, no. 1 (2007): 15–31. http://dx.doi.org/10.1017/s0004972700039435.

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A method of embedding nk configurations into projective space of k–1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a “complementary” “theorem” about projective space (over a field or skew-field F) from any nk theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues 103 (also in 3d-space), Möbius 84 (in 3d-space), and the resulting 74 in 3d-space, 96 in 5d-space, and 107 in 6d-space. (The Möbius configuration is self-complementary.) There are some nk configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.

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9

GLYNN, DAVID G. "THEOREMS OF POINTS AND PLANES IN THREE-DIMENSIONAL PROJECTIVE SPACE." Journal of the Australian Mathematical Society 88, no. 1 (2010): 75–92. http://dx.doi.org/10.1017/s1446788708080981.

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AbstractWe discuss n4 configurations of n points and n planes in three-dimensional projective space. These have four points on each plane, and four planes through each point. When the last of the 4n incidences between points and planes happens as a consequence of the preceding 4n−1 the configuration is called a ‘theorem’. Using a graph-theoretic search algorithm we find that there are two 84 and one 94 ‘theorems’. One of these 84 ‘theorems’ was already found by Möbius in 1828, while the 94 ‘theorem’ is related to Desargues’ ten-point configuration. We prove these ‘theorems’ by various methods, and connect them with other questions, such as forbidden minors in graph theory, and sets of electrons that are energy minimal.

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10

Hirasaka, Mitsugu, Kijung Kim, and Ilia Ponomarenko. "Two-valenced association schemes and the Desargues theorem." Arabian Journal of Mathematics 9, no. 3 (2019): 481–93. http://dx.doi.org/10.1007/s40065-019-00274-w.

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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.

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11

Santos, Robenilson F., Manuela S. Arruda, Ana Carla P. Bitencourt, et al. "Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations." Journal of Molecular Spectroscopy 337 (July 2017): 153–62. http://dx.doi.org/10.1016/j.jms.2017.05.005.

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12

Ivashchenko, Andrey Viktorovich, and Elena Pavlovna Znamenskaya. "FEATURES OF COMPUTER IMPLEMENTATION OF CONSTRUCTING PLANAR DESARGUES CONFIGURATION." Vestnik MGSU, no. 9 (September 2015): 168–77. http://dx.doi.org/10.22227/1997-0935.2015.9.168-177.

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The authors present the main properties of the planar configuration of Desargues, which open the possibility of its widespread use in architectural design and the design of complex volumes, consisting of a series of simple overlapping forms. However, the computer implementation of Desargues configuration construction is associated with certain difficulties caused by the fact that the monitor can only discretely represent the graphical information. In this article we identified and analyzed the properties of Desargues configuration, the use of which allows overcoming these difficulties and solving the problem of the limited capacity of monitors in the development of complex architectural forms with the help of computer graphics. Along with this, the use of the allocated properties allows predicting complex effects of the perception of architectural forms, for example, the difference of perception of architectural objects near and afar with account for perspective distortion, and they are also the basis for the development of the algorithm of construction sequence during design.

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13

Thas, C. "A rational sextic associated with a Desargues configuration." Geometriae Dedicata 51, no. 2 (1994): 163–80. http://dx.doi.org/10.1007/bf01265327.

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14

Cordovil, Raul, António Guedes de Oliveira, and Michel Las Vergnas. "A generalized Desargues configuration and the pure braid group." Discrete Mathematics 160, no. 1-3 (1996): 105–13. http://dx.doi.org/10.1016/0012-365x(95)00152-m.

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15

Ivashchenko, Andrey, and Elena Znamenskaya. "SEQUENCE VARIANTS IN THE CONSTRUCTION OF THE CONFIGURATION OF DESARGUES." Vestnik MGSU, no. 9 (September 2016): 130–39. http://dx.doi.org/10.22227/1997-0935.2016.9.130-139.

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16

Ivashchenko, A. V., and E. P. Znamenskaya. "Configuration of Desargue in architectural and design engineering." Vestnik MGSU, no. 9 (September 2014): 154–60. http://dx.doi.org/10.22227/1997-0935.2014.9.154-160.

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17

Maier, Peter, and Markus Stroppel. "Pseudo-Homogeneous Coordinates for Hughes Planes." Canadian Mathematical Bulletin 39, no. 3 (1996): 330–45. http://dx.doi.org/10.4153/cmb-1996-040-9.

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AbstractAmong the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

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18

Короткий, Виктор, and Viktor Korotkiy. "Graphic Algorithms for Constructing a Quadric, Given Nine Points." Geometry & Graphics 7, no. 2 (2019): 3–12. http://dx.doi.org/10.12737/article_5d2c1502670779.58031440.

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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.

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19

Avritzer, Dan, and Herbert Lange. "Curves of genus 2 and Desargues configurations." Advances in Geometry 2, no. 3 (2002). http://dx.doi.org/10.1515/advg.2002.013.

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20

Funk, Martin. "On the existence of little Desargues configurations in planes satisfying certain weak configurational conditions." Geometriae Dedicata 31, no. 2 (1989). http://dx.doi.org/10.1007/bf00147480.

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21

Barbut, Marc. "Sur la configuration de Desargues." Mathématiques et sciences humaines, no. 145 (March 1, 1999). http://dx.doi.org/10.4000/msh.2811.

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22

Prażmowska, Małgorzata. "MULTIPLE PERSPECTIVES AND GENERALIZATIONS OF THE DESARGUES CONFIGURATION." Demonstratio Mathematica 39, no. 4 (2006). http://dx.doi.org/10.1515/dema-2006-0418.

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23

Mendelsohn, N. S., R. Padmanabhan, and B. Wolk. "Placement of the Desargues configuration on a cubic curve." Geometriae Dedicata 40, no. 2 (1991). http://dx.doi.org/10.1007/bf00145912.

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24

OLGUN, Ş., and İ. GÜNALTILI. "On some configurational propositions connected with the minor forms of desargues." Communications, Faculty Of Science, University of Ankara Series A1Mathematics and Statistics, 1997, 153–63. http://dx.doi.org/10.1501/commua1_0000000433.

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