Hello everyone — I’m reading about the extended complex plane and the “point at infinity,” and I find the geometric picture quite helpful. According to this text, in the complex plane $z \in \mathbb{C}$, the “point at infinity” means that any sequence $\{z_n\}$ whose modulus $|z_n| \to \infty$ converges to \emph{the same} infinite point — regardless of the direction in which $z_n$ goes to infinity (i.e., along the real axis, imaginary axis, radial lines, spirals, etc.). (https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/02%3A_Analytic_Functions/2.04%3A_The_Point_at_Infinity)
Graphically this is usually described using the Riemann sphere: one can map the plane onto a sphere via stereographic projection so that “infinity” corresponds to the north pole of the sphere. (https://en.wikipedia.org/wiki/Riemann_sphere)
What I want to do (in Mathematica):
- Visualize a few sample “paths to infinity” in the complex plane — for example: radial paths, spirals, lines with different angles, or random-walk-type escapes.
- Plot their images on the Riemann sphere (via stereographic projection).
- Demonstrate visually that all these different “directions” in the plane map to \emph{the same point at infinity} (or arbitrarily close to it on the sphere).
- Optionally, show how for increasing $|z|$ along different paths, the projected points converge toward the north pole on the sphere.
My question:
Has anyone implemented such a visualization in Mathematica (or Wolfram Language)?
- Is there a built-in or user-contributed function/package that supports “plane $\to$ Riemann-sphere stereographic mapping + parametric path plotting + convergence to infinity”?
- If not, what would be a recommended approach?
- Are there pitfalls to watch out for (e.g., numerical instabilities for large $|z|$, sampling resolution, distortions near the north pole)?
Thanks in advance! Any pointers to example notebooks, snippets, or relevant resource-functions will be greatly appreciated.
