  Numerical ranges over a certain family of finite fields were classified in
2016 by a team including our fifth author. Soon afterward, in 2017 Ballico
generalized these results to all finite fields and published some new results
about the cardinality of the finite field numerical range. In this paper we
study the geometry of these finite fields using the boundary generating curve,
first introduced by Kippenhahn in 1951. We restrict our study to square
matrices of dimension 2, with at least one eigenvalue in $\mathbb F_{q^2}$.
