Blender script to perform conformal mapping from unit square to unit circle

script.py

import numpy
import bpy
import bmesh
from math import sqrt, floor
from bpy import context
from scipy.special import ellipj, ellipk

Kval = ellipk(0.5) # 1.8540746773013719

def elliptical_map(x, y):
    return x * sqrt(1.0 - y * y / 2.0), y * sqrt(1.0 - x * x / 2.0)

def cn(x, y):
    """Jacobi elliptical function cn with parameter 1/2, valid for complex numbers."""
    snx, cnx, dnx, phx = ellipj(x, 1 / 2)
    sny, cny, dny, phy = ellipj(y, 1 / 2)
    numer = cnx * cny - 1j * snx * dnx * sny * dny
    denom = (1 - dnx ** 2 * sny ** 2)
    return numer / denom

def conformal_map(x, y):
    x1 = x / 2 - y / 2
    y1 = x / 2 + y / 2
    w = cn(Kval * (1 - x1), -Kval * y1)
    u = (w.real + w.imag) / sqrt(2)
    v = (-w.real + w.imag) / sqrt(2)
    return u, v

collection = context.collection
ob = context.object
me = ob.data
sk = ob.shape_key_add(name = "Basis")
ci1 = ob.shape_key_add(name = "Elliptical")
ci2 = ob.shape_key_add(name = "Conformal")

for v in me.vertices:
    ci1.data[v.index].co.xy = elliptical_map(*v.co.xy)
    ci2.data[v.index].co.xy = conformal_map(*v.co.xy)