How to plot/visualize the inverse image of a set using computer software?

Question

Question

Let $f: \mathbb C \rightarrow S^2$ be a holomorphic map and $E\subset S^2$ be a geodesic line segment in $S^2$. I would like to plot/visualize $f^{-1}(E)\subset \mathbb C$ on the plane. What is the right software/tool to use for such a project?

Motivation

In 1997, Bowers and Stephenson introduce a "regular" pentagonal tiling of the plane (See Figure 1). I would like to construct another pentagonal tiling of the plane by "pulling back" the regular spherical pentagonal tiling of $S^2$ via a certain holomorphic map $f:\mathbb C \rightarrow S^2$.

The regular spherical pentagonal tiling of $S^2$ is the one obtained from the dodecahedron and $f$ is the square of a Weierstrass $\wp_\Lambda$ function. As the Weierstrass $\wp_\Lambda$ function is not given by an explict formula, I understand that it might be difficult to do the pullback.

Figure

Figure 1 of [BS97]

Reference

[BS97] Philip Bowers and Kenneth Stephenson. “A “regular” pentagonal tiling of the plane”. In: Conformal Geometry and Dynamics of the American Mathematical Society 1.5 (1997), pp. 58–86.