Conics from symmetric Pythagorean triple preserving matrices

Year 2019, Volume: 12 Issue: 1, 85 - 92, 27.03.2019

Mircea Crasmareanu

https://doi.org/10.36890/iejg.545845

Cited By: 2

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.

Keywords

conic, Pythagorean triple preserving matrix, complex variable

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.