code: def authalic(ell: Ellipsoid = None) -> float:
      """computes the radius of the sphere with equal surface area as the ellipsoid
  
      Parameters
      ----------
      ell : Ellipsoid, optional
            reference ellipsoid
  
      Returns
      -------
      radius: float
          radius of sphere
      """
      if ell is None:
          ell = Ellipsoid.from_name("wgs84")
  
      e = ell.eccentricity
  
      if e > 0:
          f1 = ell.semimajor_axis**2 / 2
          f2 = (1 - e**2) / (2 * e)
          f3 = log((1 + e) / (1 - e))
          return sqrt(f1 * (1 + f2 * f3))
      else:
          return ell.semimajor_axis
  
file_path: pymap3d/src/pymap3d/rsphere.py
imports: from __future__ import annotations
  from .None import rcurve
  from .ellipsoid import Ellipsoid
  from .mathfun import cos, degrees, log, radians, sin, sqrt
  from .vincenty import vdist
label_desc: Use image node_id 2 for calling a global function with example usage: authalic(ell) and returns: sqrt, ell