A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$

Year 2020, Volume: 13 Issue: 1, 21 - 40, 30.01.2020

Norbert Hungerbühler , Katharina Kusejko

https://doi.org/10.36890/iejg.590595

Abstract

We study Poncelet's Theorem in finite projective planes over the field GF(q), q = p^m for p an odd prime and m > 0, for a particular pencil of conics. We investigate whether we can find polygons with n sides which are inscribed in one conic and circumscribed around the other, so-called Poncelet Polygons. By using suitable elements of the dihedral group for these pairs, we prove that the length n of such Poncelet Polygons is independent of the starting point. In this sense Poncelet's Theorem is valid. By using Euler's divisor sum formula for the totient function, we can make a statement about the number of different conic pairs, which carry Poncelet Polygons of length n. Moreover, we will introduce polynomials whose zeros in GF(q) yield information about the relation of a given pair of conics. In particular, we can decide for a given integer n, whether and how we can find Poncelet Polygons

for pairs of conics in the plane PG(2,q). 

Keywords

Poncelet’s Theorem , finite projective planes , pencil of conics , quadratic residues

References

• [1] Abatangelo, V., Fisher, J. C., Korchmáros, G., Larato, B.: On the mutual position of two irreducible conics in PG(2; q), q odd. Adv. Geom. 11 (4), 603–614 (2011).
• [2] Berger, M.: Geometry II, Universitext. Springer-Verlag, Berlin (1987).
• [3] Bos, H. J. M., Kers, C., Oort, F., Raven, D.W.: Poncelet’s closure theorem. Exposition. Math. 5 (4), 289–364 (1987).
• [4] Cayley, A.: Developments on the porism of the in-and-circumscribed polygon. Philosophical magazine. 7 (4), 289–364 (1854).
• [5] Dragović, V., Radnović, M.: Poncelet porisms and beyond. Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel (2011).
• [6] Griffiths, P., Harris, J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign. Math. 24 (1–2), 31–40 (1978).
• [7] Halbeisen, L., Hungerbühler, N.: A Simple Proof of Poncelet’s Theorem. Amer. Math. Monthly. 122 (6), 603–614 (2015).
• [8] Hardy, G. H., Wright, E. M.: An introduction to the theory of numbers. Oxford University Press, Oxford (2008).
• [9] Hirschfeld, J.W. P.: Projective geometries over finite fields. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1998).
• [10] Hungerbühler, N., Kusejko, K.: Poncelet’s Theorem in the four non-isomorphic finite projective planes of order 9. Ars Combin. 140, 21–44 (2018).
• [11] Korchmáros, G., Sz˝onyi, T.: Affinely regular polygons in an affine plane. Contrib. Discrete Math. 3 (1), 20–38 (2008).
• [12] Kusejko, K.: Simultaneous diagonalization of conics in PG(2; q). Des. Codes Cryprogr. 79 (3), 565–581 (2016).
• [13] Luisi, G.: On a theorem of Poncelet. Atti Sem. Mat. Fis. Univ. Modena. 31 (2), 341–347 (1984).
• [14] Poncelet, J.-V.: Traité des propriétés projectives des figures. Tome II. Les Grands Classiques Gauthier-Villars. Reprint of the second (1866) edition. Éditions Jacques Gabay, Sceaux (1995).