Calculations of Dihedral Groups Using Circular Indexation

Document Type: Special Issue: International Conference on Architecture and Mathematics

Authors

1 Department of Electrical Engineering, Persian Gulf University

2 Department of Pure Mathematics, Persian Gulf University

Abstract

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and generates a processed circular output sequence‎. ‎The system can be described by a permutation function‎. ‎Permutation functions can be written in a simple form using circular indexation‎. ‎The operation between the symmetries of the polygon is reduced to the composition of permutation functions‎, ‎which in turn is easily implemented using periodic sequences‎. ‎It is also shown that each symmetry is effectively a pure rotation or a pure flip‎. ‎It is also explained how to synthesize each symmetry using two generating symmetries‎: ‎time-reversal (flipping around a fixed symmetry axis) and unit-delay (rotation around the center-point by $2\pi‎ /‎n$ radians clockwise)‎. ‎The group of the symmetries of a polygon is called a dihedral group and it has applications in different engineering fields including image processing‎, ‎error correction codes in telecommunication engineering‎, ‎remote sensing, and radar‎.

Keywords


[1] N. S. Akbar, Mathematical model for blood flow through tapered arteries with
temperature dependent viscosity, J. Adv. Math. Appl. 3 (2014) 122 − 129.
[2] G. Arfken, Mathematical Methods for Physicists, 3rd Ed., Orlando: FL: Aca-
demic Press, New York-London, 1985.
[3] R. S. Caprari, Geometric symmetry in the quadratic Fisher discriminant op-
erating on image pixels, IEEE Trans. Inform. Theory 52 (2006) 1780−1788.
[4] T. L. Kunii, H. Nishida and M. Hilaga, Topological modeling for visualization,
J. Adv. Math. Appl. 1 (2012) 134 − 150.
[5] R. Lenz, Using representations of the dihedral groups in the design of early
vision filters, In: Proc. International Conference on Acoustics, Speech, and
Signal Processing, ICASSP-93, Minneapolis, MN, USA, 1993.
[6] G. Mayhew, Group property of the P4, K4, and NR16 error correction codes,
In: Proc. Aerospace Conference, 2017.
[7] A. V. Oppenheim, R. W. Schafer and J. R. Buck Discrete-Time Signal Processing
, 2nd ed, Prentice-Hall Signal Processing Series, Pearson, Upper Saddle
River, N. J., 1998.

[8] D. Saracino, Abstract Algebra: A First Course, Addison Wesley Longman
Publishing Co, USA, 1980.
[9] A. Thiele, E. Cadario, K. Schulz, U. Thoennessen and U. Soergel, Feature
extraction of gable-roofed buildings from multi-aspect high-resolution InSAR
data, In: Proc. International Geoscience and Remote Sensing Symposium,
(2007) 262 − 265.
[10] M. Thill and B. Hassibi, Frames from groups: Generalized bounds and di-
hedral groups, In: Proc. International Conference on Acoustics, Speech and
Signal Processing, (2013) 6043 − 6047.
[11] Y. Wang, W. Pedrycz, J. Lu and G. Luo, Denotational mathematical models
of an air traffic control system (ATCS-II): Process models of functions in
RTPA, J. Adv. Math. Appl. 2 (2013) 82 − 110.