$C$ can be recovered by taking the inverse image of this set under the mapping $g$, that is $C = \{(x + iy)^{-(4-d)/8}\, |\, xy = A_q,\, x \in I_x\}$ where $I_x$ is an interval such that $(x+iy)^{-(4-d)/8} \in B(a,r_0)$ for $x \in I_x$.

provides a simple method for computing the trajectory of a streamline through a singular triangle. We make the assumption that within the triangle, the estimate $$f(z) \approx e^{i\left(\frac{d\theta}{4} + \frac{2k\pi}{4}\right)}$$

holds. Here again $\theta = 0$ corresponds to the nearest separatrix clockwise from $q$. Making this assumption, we simply compute points along the hyperbola $xy = A_q$, and take the inverse image of each point. We use these points as discretization points of the streamline so long as they lay within the singular triangle. Since hyperbolas are convex, and $g^{-1}$ preserves the order of points along rays, in order to guarantee that two streamlines don’t intersect tangentially, it is sufficient to evaluate the points of the hyperbola along predefined rays from the singular point.

Partition Construction and Tangential Crossings {#sec:tangential-crossings}
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By the Poincaré-Bendixson theorem for manifolds [@schwartz_generalization_1963], streamlines of a cross field on a bounded manifold $M$ can do one of the following:

1.  Connect 