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Conics from symmetric Pythagorean triple preserving matrices

Year 2019, , 85 - 92, 27.03.2019
https://doi.org/10.36890/iejg.545845

Abstract

The aim of this paper is to introduce and study the class of conics provided by symmetric
Pythagorean triple preserving matrices. This class depends on three real parameters and various
relationships between these parameters give special subclasses. A symmetric matrix of Barning
and its associated hyperbola are carefully analyzed through a pair of points of rational coordinates.
We transform and extend this latter hyperbola to a class of hyperbolas containing integral points
(k; k + 3), (k + 3; k). A complex approach is also included.

References

  • [1] Austin, H. W. and Austin, J. W., On a special set of symmetric Pythagorean triple preserving matrices. Adv. Appl. Math. Sci. 12 (2012), no. 2, 97-104.
  • [2] Barning, F. J. M., On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices (Dutch). Math. Centrum, Amsterdam, Afd. Zuivere Wisk. ZW, 1963-011, 37 p.
  • [3] Bruening, J. T., Lohmeier, T. R. and Sebaugh, C. L., Symmetric Pythagorean triple preserving matrices. Missouri J. Math. Sci. 13 (2001), no. 1, 4-16.
  • [4] Crasmareanu, M., A new method to obtain Pythagorean triple preserving matrices. Missouri J. Math. Sci. 14 (2002), no. 3, 149-158.
  • [5] Crasmareanu, M., A complex approach to the gradient-type deformations of conics. Bull. Transilv. Univ. Bra¸sov, Ser. III, Math. Inform. Phys. 10(59) (2017), no. 2, 59-62.
  • [6] Katayama, Shin-ichi, Modified Farey trees and Pythagorean triples. J. Math., Univ. Tokushima 47 (2013), 1-13.
  • [7] Murasaki, T., On the Heronian triple (n + 1; n; n - 1) (Japanese). Sci. Rep. Fac. Educ. Gunma Univ. 52 (2004), 9-15.
  • [8] Murru, N., Abrate, M., Barbero, S. and Cerruti, U., Groups and monoids of Pythagorean triples connected to conics. Open Math. 15 (2017), 1323-1331.
  • [9] Palmer, L., Ahuja, M. and Tikoo, M., Finding Pythagorean triple preserving matrices. Missouri J. Math. Sci. 10 (1998), no. 2, 99-105.
  • [10] Palmer, L., Ahuja, M. and Tikoo, M., Constructing Pythagorean triple preserving matrices. Missouri J. Math. Sci. 10 (1998), no. 3, 159-168.
  • [11] Romik, D., The dynamics of Pythagorean triples. Trans. Am. Math. Soc. 360 (2008), no. 11, 6045-6064.
  • [12] Tikoo, M., A note on Pythagorean triple preserving matrices. Internat. J. Math. Ed. Sci. Tech. 33 (2002), no. 6, 893-894.
  • [13] Tikoo, M. andWang, H., Generalized Pythagorean triples and Pythagorean triple preserving matrices. Missouri J. Math. Sci. 21 (2009), no. 1, 3-12.
Year 2019, , 85 - 92, 27.03.2019
https://doi.org/10.36890/iejg.545845

Abstract

References

  • [1] Austin, H. W. and Austin, J. W., On a special set of symmetric Pythagorean triple preserving matrices. Adv. Appl. Math. Sci. 12 (2012), no. 2, 97-104.
  • [2] Barning, F. J. M., On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices (Dutch). Math. Centrum, Amsterdam, Afd. Zuivere Wisk. ZW, 1963-011, 37 p.
  • [3] Bruening, J. T., Lohmeier, T. R. and Sebaugh, C. L., Symmetric Pythagorean triple preserving matrices. Missouri J. Math. Sci. 13 (2001), no. 1, 4-16.
  • [4] Crasmareanu, M., A new method to obtain Pythagorean triple preserving matrices. Missouri J. Math. Sci. 14 (2002), no. 3, 149-158.
  • [5] Crasmareanu, M., A complex approach to the gradient-type deformations of conics. Bull. Transilv. Univ. Bra¸sov, Ser. III, Math. Inform. Phys. 10(59) (2017), no. 2, 59-62.
  • [6] Katayama, Shin-ichi, Modified Farey trees and Pythagorean triples. J. Math., Univ. Tokushima 47 (2013), 1-13.
  • [7] Murasaki, T., On the Heronian triple (n + 1; n; n - 1) (Japanese). Sci. Rep. Fac. Educ. Gunma Univ. 52 (2004), 9-15.
  • [8] Murru, N., Abrate, M., Barbero, S. and Cerruti, U., Groups and monoids of Pythagorean triples connected to conics. Open Math. 15 (2017), 1323-1331.
  • [9] Palmer, L., Ahuja, M. and Tikoo, M., Finding Pythagorean triple preserving matrices. Missouri J. Math. Sci. 10 (1998), no. 2, 99-105.
  • [10] Palmer, L., Ahuja, M. and Tikoo, M., Constructing Pythagorean triple preserving matrices. Missouri J. Math. Sci. 10 (1998), no. 3, 159-168.
  • [11] Romik, D., The dynamics of Pythagorean triples. Trans. Am. Math. Soc. 360 (2008), no. 11, 6045-6064.
  • [12] Tikoo, M., A note on Pythagorean triple preserving matrices. Internat. J. Math. Ed. Sci. Tech. 33 (2002), no. 6, 893-894.
  • [13] Tikoo, M. andWang, H., Generalized Pythagorean triples and Pythagorean triple preserving matrices. Missouri J. Math. Sci. 21 (2009), no. 1, 3-12.
There are 13 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Mircea Crasmareanu

Publication Date March 27, 2019
Published in Issue Year 2019

Cite

APA Crasmareanu, M. (2019). Conics from symmetric Pythagorean triple preserving matrices. International Electronic Journal of Geometry, 12(1), 85-92. https://doi.org/10.36890/iejg.545845
AMA Crasmareanu M. Conics from symmetric Pythagorean triple preserving matrices. Int. Electron. J. Geom. March 2019;12(1):85-92. doi:10.36890/iejg.545845
Chicago Crasmareanu, Mircea. “Conics from Symmetric Pythagorean Triple Preserving Matrices”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 85-92. https://doi.org/10.36890/iejg.545845.
EndNote Crasmareanu M (March 1, 2019) Conics from symmetric Pythagorean triple preserving matrices. International Electronic Journal of Geometry 12 1 85–92.
IEEE M. Crasmareanu, “Conics from symmetric Pythagorean triple preserving matrices”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 85–92, 2019, doi: 10.36890/iejg.545845.
ISNAD Crasmareanu, Mircea. “Conics from Symmetric Pythagorean Triple Preserving Matrices”. International Electronic Journal of Geometry 12/1 (March 2019), 85-92. https://doi.org/10.36890/iejg.545845.
JAMA Crasmareanu M. Conics from symmetric Pythagorean triple preserving matrices. Int. Electron. J. Geom. 2019;12:85–92.
MLA Crasmareanu, Mircea. “Conics from Symmetric Pythagorean Triple Preserving Matrices”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 85-92, doi:10.36890/iejg.545845.
Vancouver Crasmareanu M. Conics from symmetric Pythagorean triple preserving matrices. Int. Electron. J. Geom. 2019;12(1):85-92.