Trigonometric-Type Functions Derived from Polygons Inscribed in the Unit Circle

Abstract

Given a polygon inscribed inside of a circle with vertices satisfying the equation z ^N = 1, we introduce a new class of periodic functions called the “geometric polygon functions.” Methodology used to construct and analyze the classical circular and elliptic functions is essential for defining the cosine polygon, sine polygon, and dine polygon functions, called the “geometric polygon functions.” Dividing N into two sub-cases, odd values and even values, allows mutually exclusive sets. Focusing on even values of N, the square functions are introduced as the smallest most significant case within this sub-case. The square functions provide the building blocks for larger even values of N for which additional analysis is presented. The goals of this research are to introduce the “geometric polygon functions” and compute the Fourier series expansion of the square functions. These findings can be manipulated to prove results in matroid theory. In particular, the construct of the “geometric polygon functions” are representations of graphs whose bicircular matroids have well controlled circuit spectra.