method to accurately trace streamlines within the neighborhood of a singularity to any predefined resolution. This method guarantees that no tangential intersections of streamlines will occur within the neighborhood. We first prove that in $\mathbb{R}^2$, trajectories of streamlines through the neighborhood of a singularity are hyperbolas under a conformal transformation.

Let $f$ be a canonical harmonic cross field (see [@viertel_approach_2019]) on a domain $D \subset {\mathbb R}^2$. Let $a \in {\mathbb R}^2$ be the location of a singularity of index $\frac{d}{4}$ where $d$ is an integer $\leq 1$. Consider the open ball $B(a,r_0)$ of radius $r_0 > 0$ centered at $a$. We seek an approximation for the trajectory of an arbitrary streamline passing through a point $q \neq a$ in $B(a,r_0)$.

The cross field, $f(z)$, partitions $B(a,r_0)$ into $4 - d$ evenly angled sectors bounded by separatrices of the cross field [@viertel_approach_2019]. In each sector, the cross field defines a local $(u,v)$ parameterization. Let $S$ be the open sector containing $q$, and let $s_0$ and $s_1$ be the separatrices bounding $S$ ordered counterclockwise. Because the cross field defines a local $(u,v)$ parameterization on $S$, there are two streamlines passing through $q$, one crossing $s_0$ orthogonally, the other crossing $s_1$ orthogonally. 