Secant Distributions of Unitals

Let U be a unital embedded in a projective plane \(\Pi \) $\mathrm{\Pi }$ of order \(q^2\) $q^2$ . For \(R\in U\) $R\in U$ , let \(s_R\) $s_R$ and \(t_R\) $t_R$ be a secant line through R and the tangent line to U at point R, respectively. If the tangent lines to U, passing through the points in \(s_R\cap U\) $s_R\cap U$ , intersect at a single point on \(t_R\) $t_R$ , then \(s_R\) $s_R$ is referred to as a secant line satisfying the desired property. If \(n_i\) $n_i$ of the points of U have exactly \(m_i\) $m_i$ secant lines satisfying the desired property, then \(\begin{aligned} m_1^{n_1}, m_2^{n_2}, \cdots \end{aligned}\) $$\begin{array}{c}\hfill m_1^{n_1},m_2^{n_2},\cdots \end{array}$$ is called the secant distribution of U, where \(\sum n_i=q^3+1\) $\sum n_i=q^3+1$ , and \(0\le m_i\le q^2\) $0\le m_i\le q^2$ . In this article, we show that collinear pedal sets of a unital U plays an important role in the secant distribution of U. Formulas for secant distributions of unitals having \(0,1,q^2,\) $0,1,q^2,$ or \(q^2+q\) $q^2+q$ special points are provided. Statistics regarding to secant distributions of unitals embedded in planes of orders \(q^2\le 25\) $q^2\le 25$ are presented. Some open problems related to secant distributions of unitals having specific number of collinear pedal sets are discussed.