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Journal articles on the topic 'Geometric algebra for conics'

Author: Grafiati

Published: 28 June 2021

Last updated: 3 February 2022

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1

Dragovic, Vladimir. "Algebro-geometric approach to the Yang-Baxter equation and related topics." Publications de l'Institut Math?matique (Belgrade) 91, no. 105 (2012): 25–48. http://dx.doi.org/10.2298/pim1205025d.

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We review the results of algebro-geometric approach to 4 ? 4 solutions of the Yang-Baxter equation. We emphasis some further geometric properties, connected with the double-reflection theorem, the Poncelet porism and the Euler-Chasles correspondence. We present a list of classifications in Mathematical Physics with a similar geometric background, related to pencils of conics. In the conclusion, we introduce a notion of discriminantly factorizable polynomials as a result of a computational experiment with elementary n-valued groups.

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2

Halbeisen, Lorenz, and Norbert Hungerbühler. "The exponential pencil of conics." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 59, no. 3 (December 21, 2017): 549–71. http://dx.doi.org/10.1007/s13366-017-0375-1.

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3

RASHED, ROSHDI. "LES CONSTRUCTIONS GÉOMÉTRIQUES ENTRE GÉOMÉTRIE ET ALGÈBRE: L'ÉPÎTRE D'AB AL-JD À AL-BRN." Arabic Sciences and Philosophy 20, no. 1 (March 2010): 1–51. http://dx.doi.org/10.1017/s0957423909990075.

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AbstractAbū al-Jūd Muḥammad ibn al-Layth is one of the mathematicians of the 10th century who contributed most to the novel chapter on the geometric construction of the problems of solids and super-solids, and also to another chapter on solving cubic and bi-quadratic equations with the aid of conics. His works, which were significant in terms of the results they contained, are moreover important with regard to the new relations they established between algebra and geometry. Good fortune transmitted to us his correspondences with the mathematician and astronomer al-Bīrūnī. The questions they debated, and the answers they yielded, all offer us multiple in vivo perspectives on the research that was undertaken in that period. The reader would find in this article a critical edition and French translation of this correspondence, with historical and mathematical commentaries.

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4

Halbeisen, Lorenz, and Norbert Hungerbühler. "Closed chains of conics carrying poncelet triangles." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 58, no. 2 (January 18, 2017): 277–302. http://dx.doi.org/10.1007/s13366-016-0327-1.

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5

Halbeisen, Lorenz, and Norbert Hungerbühler. "Generalized pencils of conics derived from cubics." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 61, no. 4 (April 15, 2020): 681–93. http://dx.doi.org/10.1007/s13366-020-00499-3.

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6

Mirman, Boris. "Short cycles of Poncelet’s conics." Linear Algebra and its Applications 432, no. 10 (May 2010): 2543–64. http://dx.doi.org/10.1016/j.laa.2009.11.032.

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7

Diemente, Damon. "Algebra in the Service of Geometry: Can Euler's Line Be Parallel to a Side of a Triangle?" Mathematics Teacher 93, no. 5 (May 2000): 428–31. http://dx.doi.org/10.5951/mt.93.5.0428.

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This investigation of Euler's line has become a regular and valued unit in my honors–geometry syllabus. It originated with an intelligent question from a curious student. Its geometric foundation comprises sophisticated Euclidean triangle geometry. Its solution requires plentiful but not excessively complicated algebra. It culminates in the discovery of a conic locus that can be verified by construction on a computer screen.

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8

Nievergelt, Yves. "Fitting conics of specific types to data." Linear Algebra and its Applications 378 (February 2004): 1–30. http://dx.doi.org/10.1016/j.laa.2003.08.022.

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9

Wu, Junhua. "Conics arising from internal points and their binary codes." Linear Algebra and its Applications 439, no. 2 (July 2013): 422–34. http://dx.doi.org/10.1016/j.laa.2013.04.004.

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10

Easter, Robert Benjamin, and Eckhard Hitzer. "Conic and cyclidic sections in double conformal geometric algebra G8,2 with computing and visualization using Gaalop." Mathematical Methods in the Applied Sciences 43, no. 1 (September 9, 2019): 334–57. http://dx.doi.org/10.1002/mma.5887.

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11

Zhang, Ying, Xin Liu, Shimin Wei, Yaobing Wang, Xiaodong Zhang, Pei Zhang, and Changchun Liang. "A Geometric Modeling and Computing Method for Direct Kinematic Analysis of 6-4 Stewart Platforms." Mathematical Problems in Engineering 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/6245341.

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A geometric modeling and solution procedure for direct kinematic analysis of 6-4 Stewart platforms with any link parameters is proposed based on conformal geometric algebra (CGA). Firstly, the positions of the two single spherical joints on the moving platform are formulated by the intersection, dissection, and dual of the basic entities under the frame of CGA. Secondly, a coordinate-invariant equation is derived via CGA operation in the positions of the other two pairwise spherical joints. Thirdly, the other five equations are formulated in terms of geometric constraints. Fourthly, a 32-degree univariate polynomial equation is reduced from a constructed 7 by 7 matrix which is relatively small in size by using a Gröbner-Sylvester hybrid method. Finally, a numerical example is employed to verify the solution procedure. The novelty of the paper lies in that (1) the formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA and (2) the size of the resultant matrix is smaller than those existed.

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12

Gao, Xiao-Shan, Kun Jiang, and Chang-Cai Zhu. "Geometric constraint solving with conics and linkages." Computer-Aided Design 34, no. 6 (May 2002): 421–33. http://dx.doi.org/10.1016/s0010-4485(01)00114-2.

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13

Cantón, A., L. Fernández-Jambrina, and E. Rosado María. "Geometric characteristics of conics in Bézier form." Computer-Aided Design 43, no. 11 (November 2011): 1413–21. http://dx.doi.org/10.1016/j.cad.2011.08.025.

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14

GOLDMAN, RON, and WENPING WANG. "USING INVARIANTS TO EXTRACT GEOMETRIC CHARACTERISTICS OF CONIC SECTIONS FROM RATIONAL QUADRATIC PARAMETERIZATIONS." International Journal of Computational Geometry & Applications 14, no. 03 (June 2004): 161–87. http://dx.doi.org/10.1142/s021819590400141x.

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Extracting the geometric characteristics of conic sections, such as their center, axes and foci, from their defining equations is required for various applications in computer graphics and geometric modeling. Although there exist standard techniques for computing the geometric characteristics for conics in implicit form, in shape modeling applications conic sections are often represented by rational quadratic parameterizations. Here we present closed formulas for computing the geometric characteristics of conics directly from their quadratic parameterizations without resorting to implicitization procedures. Our approach uses the invariants of rational quadratic parameterizations under rational linear reparameterizations. These invariants are also used to give a complete characterization of degenerate tonics represented by rational quadratic parameterizations.

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15

Girsh, A. "Dual Problems with Conics." Geometry & Graphics 8, no. 1 (April 20, 2020): 15–24. http://dx.doi.org/10.12737/2308-4898-2020-15-24.

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The problem for construction of straight lines, which are tangent to conics, is among the dual problems for constructing the common elements of two conics. For example, the problem for construction of a chordal straight line (a common chord for two conics) ~ the problem for construction of an intersection point for two conics’ common tangents. In this paper a new property of polar lines has been presented, constructive connection between polar lines and chordal straight lines has been indicated, and a new way for construction of two conics’ common chords has been given, taking into account the computer graphics possibilities. The construction of imaginary tangent lines to conic, traced from conic’s interior point, as well as the construction of common imaginary tangent lines to two conics, of which one lies inside another partially or thoroughly is considered. As you know, dual problems with two conics can be solved by converting them into two circles, followed by a reverse transition from the circles to the original conics. This method of solution provided some clarity in understanding the solution result. The procedure for transition from two conics to two circles then became itself the subject of research. As and when the methods for solving geometric problems is improved, the problems themselves are become more complex. When assuming the participation of imaginary images in complex geometry, it is necessary to abstract more and more. In this case, the perception of the obtained result’s geometric picture is exposed to difficulties. In this regard, the solution methods’ correctness and imaginary images’ visualization are becoming relevant. The paper’s main results have been illustrated by the example of the same pair of conics: a parabola and a circle. Other pairs of affine different conics (ellipse and hyperbola) have been considered in the paper as well in order to demonstrate the general properties of conics, appearing in investigated operations. Has been used a model of complex figures, incorporating two superimposed planes: the Euclidean plane for real figures, and the pseudo-Euclidean plane for imaginary algebraic figures and their imaginary complements.

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16

UNEL, MUSTAFA, and WILLIAM A. WOLOVICH. "A NEW REPRESENTATION FOR QUARTIC CURVES AND COMPLETE SETS OF GEOMETRIC INVARIANTS." International Journal of Pattern Recognition and Artificial Intelligence 13, no. 08 (December 1999): 1137–49. http://dx.doi.org/10.1142/s0218001499000641.

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Many free-form object boundaries can be modeled by quartics with bounded zero sets. The fact that any nondegenerate closed-bounded algebraic curve of even degree n=2p can be expressed as the product of p conics, which are real ellipses, plus a remaining polynomial of degree n-2,12 can be utilized to express a nondegenerate quartic as the product of two leading ellipses plus a third conic which might be either a closed curve (an ellipse) or an open curve (a hyperbola). However, it can be shown that the leading ellipses can be modified with appropriate constants by constraining the third conic to be a circle, thus implying a 2-ellipse and 1-circle; i.e. an elliptical-circular(E2C)representation of the quartic. The use of such representations is to simplify the analysis of quartics by exploiting the well-known properties of conics and to develop a set of functionally independent geometric invariants for recognition purposes. Also, it is shown that the underlying Euclidean transformation between two configurations of the same quartic can be determined using the centers of the three conics.

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17

Kopacz, Piotr. "ERRATUM." Geodesy and cartography 40, no. 2 (June 24, 2014): EBI. http://dx.doi.org/10.3846/20296991.2013.763633.

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18

van Hoeij, Mark, and John Cremona. "Solving conics over function fields." Journal de Théorie des Nombres de Bordeaux 18, no. 3 (2006): 595–606. http://dx.doi.org/10.5802/jtnb.560.

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19

Sobczyk, Garret. "Geometric matrix algebra." Linear Algebra and its Applications 429, no. 5-6 (September 2008): 1163–73. http://dx.doi.org/10.1016/j.laa.2007.06.015.

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20

Sobczyk, Garret. "Unitary Geometric Algebra." Advances in Applied Clifford Algebras 22, no. 3 (July 18, 2012): 827–36. http://dx.doi.org/10.1007/s00006-012-0364-2.

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21

Caravantes, Jorge, Gema M. Diaz-Toca, Mario Fioravanti, and Laureano Gonzalez-Vega. "On the Implicit Equation of Conics and Quadrics Offsets." Mathematics 9, no. 15 (July 28, 2021): 1784. http://dx.doi.org/10.3390/math9151784.

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A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics.

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22

Klawitter, Daniel. "Reflections in conics, quadrics and hyperquadrics via Clifford algebra." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 57, no. 1 (September 10, 2014): 221–42. http://dx.doi.org/10.1007/s13366-014-0218-2.

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23

Brundu, Michela, and Gianni Sacchiero. "On Rational Surfaces Ruled by Conics." Communications in Algebra 31, no. 8 (January 9, 2003): 3631–52. http://dx.doi.org/10.1081/agb-120022436.

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24

Sugon, Quirino M., and Daniel J. McNamara. "A geometric algebra reformulation of geometric optics." American Journal of Physics 72, no. 1 (January 2004): 92–97. http://dx.doi.org/10.1119/1.1621029.

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25

Belon, Mauricio Cele Lopez, and Dietmar Hildenbrand. "Practical Geometric Modeling Using Geometric Algebra Motors." Advances in Applied Clifford Algebras 27, no. 3 (April 1, 2017): 2019–33. http://dx.doi.org/10.1007/s00006-017-0777-z.

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26

Sánchez-Reyes, Javier, and Marco Paluszny. "Weighted radial displacement: A geometric look at Bézier conics and quadrics." Computer Aided Geometric Design 17, no. 3 (March 2000): 267–89. http://dx.doi.org/10.1016/s0167-8396(99)00051-5.

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27

Sánchez-Reyes, J. "Geometric recipes for constructing Bézier conics of given centre or focus." Computer Aided Geometric Design 21, no. 2 (February 2004): 111–16. http://dx.doi.org/10.1016/j.cagd.2003.09.001.

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28

Sánchez-Reyes, J. "Simple determination via complex arithmetic of geometric characteristics of Bézier conics." Computer Aided Geometric Design 28, no. 6 (August 2011): 345–48. http://dx.doi.org/10.1016/j.cagd.2011.06.007.

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29

Vincze, Cs, and Á. Nagy. "On the theory of generalized conics with applications in geometric tomography." Journal of Approximation Theory 164, no. 3 (March 2012): 371–90. http://dx.doi.org/10.1016/j.jat.2011.11.004.

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30

Malyshev, V. A. "Poncelet problem for rational conics." St. Petersburg Mathematical Journal 19, no. 4 (May 9, 2008): 597–601. http://dx.doi.org/10.1090/s1061-0022-08-01012-1.

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31

Kopacz, Piotr. "ON GEOMETRIC PROPERTIES OF SPHERICAL CONICS AND GENERALIZATION OF Π IN NAVIGATION AND MAPPING." Geodesy and Cartography 38, no. 4 (December 21, 2012): 141–51. http://dx.doi.org/10.3846/20296991.2012.756995.

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First, we cover the conical curves on 2-dimensional modeling sphere S 2 showing their geometric properties affecting the hyperbolic navigation. We place emphasis on the geometric definition of spherical parabola and relate it to the notions of spherical ellipse and hyperbola and give simple geometric proofs for relations between conical curves on the sphere. In the second part of the paper function representing the ratio of the circle's circumference to its diameter has been defined and researched to analyze the potential discrepancies in the spherical and conical projective models on which the navigational computations are based on. We compare some non-Euclidean geometric properties of curved surfaces and its Euclidean plane model in reference to the local and global approximation. As a working tool we use function for geometric comparison analysis in the theory of long-range navigation and cartographic projection. We state the existence of the infinite number of the circles having the same radius but different circumference on the conical surface. Finally, we survey the exemplary proposals of generalization of function . In particular, we focus on the geometric structure of applied model treated as a metric space showing the differences in the outputting computations if the changes in a metric are made. We also relate the function to Tissot's indicatrix of distortion.

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32

Druzhinin, V. V. "ALGEBRA OF GEOMETRIC PROGRESSIONS." Scientific and Technical Volga region Bulletin 7, no. 1 (February 2017): 18–20. http://dx.doi.org/10.24153/2079-5920-2017-7-1-18-20.

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33

Lopes, Wilder Bezerra, and Cassio Guimaraes Lopes. "Geometric-Algebra Adaptive Filters." IEEE Transactions on Signal Processing 67, no. 14 (July 15, 2019): 3649–62. http://dx.doi.org/10.1109/tsp.2019.2916028.

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34

Terese, Brittany, and David L. Millman. "Review of Geometric algebra." ACM SIGACT News 42, no. 1 (March 21, 2011): 46–48. http://dx.doi.org/10.1145/1959045.1959057.

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35

Arcaute, Elsa, Anthony Lasenby, and Chris Doran. "Twistors in Geometric Algebra." Advances in Applied Clifford Algebras 18, no. 3-4 (May 19, 2008): 373–94. http://dx.doi.org/10.1007/s00006-008-0083-x.

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36

Cameron, Jonathan, and Joan Lasenby. "Oriented Conformal Geometric Algebra." Advances in Applied Clifford Algebras 18, no. 3-4 (May 19, 2008): 523–38. http://dx.doi.org/10.1007/s00006-008-0084-9.

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37

Easter, Robert Benjamin, and Eckhard Hitzer. "Double Conformal Geometric Algebra." Advances in Applied Clifford Algebras 27, no. 3 (April 20, 2017): 2175–99. http://dx.doi.org/10.1007/s00006-017-0784-0.

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38

Petitjean, Michel. "Chirality in Geometric Algebra." Mathematics 9, no. 13 (June 29, 2021): 1521. http://dx.doi.org/10.3390/math9131521.

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We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used.

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39

Strickland, E. "The equivariant ring of conditions of conics." Journal of Algebra 329, no. 1 (March 2011): 274–85. http://dx.doi.org/10.1016/j.jalgebra.2009.12.002.

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40

Hajja, M., M. C. Kang, and J. Ohm. "Function Fields of Conics as Invariant Subfields." Journal of Algebra 163, no. 2 (January 1994): 383–403. http://dx.doi.org/10.1006/jabr.1994.1024.

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41

Cremona, J. E., and D. Rusin. "Efficient solution of rational conics." Mathematics of Computation 72, no. 243 (December 18, 2002): 1417–42. http://dx.doi.org/10.1090/s0025-5718-02-01480-1.

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42

MILED, MAROUANE BEN. "MESURER LE CONTINU, DANS LA TRADITION ARABE DES LIVRES V ET X DES ÉLÉMENTS." Arabic Sciences and Philosophy 18, no. 1 (March 2008): 1–18. http://dx.doi.org/10.1017/s0957423908000453.

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In order to find positive solutions for third-degree equations, which he did not know how to solve for roots, ‘Umar al-Khayyām proceeds by the intersections of conic sections. The representation of an algebraic equation by a geometrical curve is made possible by the choices of units of measure for lengths, surfaces, and volumes. These units allow a numerical quantity to be associated with a geometrical magnitude. Is there a trace of this unit in the mathematicians to whom al-Khayyām refers directly in his Algebra? How does this unit enable the measurement of quantities and rational and irrational relations? We find answers to these questions in the commentaries to Books V and X of the Elements.

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43

Macdonald, Alan. "A Survey of Geometric Algebra and Geometric Calculus." Advances in Applied Clifford Algebras 27, no. 1 (April 25, 2016): 853–91. http://dx.doi.org/10.1007/s00006-016-0665-y.

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44

Hongbo, Li. "Geometric interpretations of gradednull monomials inconformal geometric algebra." SCIENTIA SINICA Mathematica 51, no. 1 (August 10, 2020): 179. http://dx.doi.org/10.1360/ssm-2019-0329.

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45

Parimala, R. "Witt groups of conics, elliptic, and hyperelliptic curves." Journal of Number Theory 28, no. 1 (January 1988): 69–93. http://dx.doi.org/10.1016/0022-314x(88)90120-5.

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46

Olmstead, Eugene A., and Arne Engebretsen. "Technology Tips: Exploring the Locus Definitions of the Conic Sections." Mathematics Teacher 91, no. 5 (May 1998): 428–34. http://dx.doi.org/10.5951/mt.91.5.0428.

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Conic sections were first studied in 350 B.C. by Menaechmus, who cut a circular conical surface at various angles. Early mathematicians who added to the study of conics include Apollonius, who named them in 220 B.C., and Archimedes, who studied their fascinating properties around 212 B.C. In previous articles in this journal, conic sections have been shown both as an algebraic, or parametric, representation (Vonder Embse 1997) and as a geometric, that is, a paper-folding, model (Scher 1996). Both articles offer important insights into the mathematical nature of the conic sections and into teaching methods that can integrate conics into our curriculum. Even though many textbooks discuss conic equations and their graphs, they do not fully develop locus definitions of conic sections.

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47

Shiu, P., E. Sernesi, and J. Montaldi. "Linear Algebra; A Geometric Approach." Mathematical Gazette 79, no. 484 (March 1995): 207. http://dx.doi.org/10.2307/3620085.

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48

Kilmister, C. W., D. Hestenes, and G. Sobczyk. "Clifford Algebra to Geometric Calculus." Mathematical Gazette 69, no. 448 (June 1985): 158. http://dx.doi.org/10.2307/3616966.

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49

Hestenes, David. "Spacetime physics with geometric algebra." American Journal of Physics 71, no. 7 (July 2003): 691–714. http://dx.doi.org/10.1119/1.1571836.

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50

RESENDES, D. P. "Geometric algebra in plasma electrodynamics." Journal of Plasma Physics 79, no. 5 (April 12, 2013): 735–38. http://dx.doi.org/10.1017/s0022377813000366.

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AbstractGeometric algebra (GA) is a recent broad mathematical framework incorporating synthetic and coordinate geometry, complex variables, quarternions, vector analysis, matrix algebra, spinors, tensors, and differential forms. It has been claimed to be a unified language for physics. GA is presented in the context of the Maxwell-Plasma system. In this formalism the divergence and curl differential operators are united in a single vector derivative, which is invertible, in the form of a first-order Green function. The four Maxwell equations can be combined into a single equation (for homogeneous and constant media) or into two equations involving the invertible vector derivative for more complex media. GA is applied to simple examples to illustrate the compactness of the notation and coordinate-free computations.

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