A sketch of an algorithm for the Steinitz representation for a planar graph

Our goal is to arrange vertices so that each vertex is associated with a circle on a sphere in a way that two vertices are adjacent if and only if the associated circles are tangent to each other.

(An alternative description of the Steinitz representation is to represent the graph as a polytope that wraps around a sphere in a way that every edge is tangent to the sphere.)

A picture of a Steinitz representation

An outline of the algorithm

Input: a drawing of a planar graph in the plane
Output: computing the radius of discs associated with vertices and the radius of the sphere.
- Triangulate the planar drawing by adding edges if necessary. The resulting drawing has all faces being triangles.
- Step 0: Initialization:
  - For each vertex v, do the following:
    If its degree d_v is at most 5, assign weight 1/2 to each vertex v i.e., set
    r_i = 1/2.
    Otherwise, set
    r_i = d_v/10.
  - Compute R_v (i.e., the radius of the ball that fits at the cone formed at v).
  - Set R = average R_v.
- Step i:
  Compute, for each vertex v, angles at v in faces incident to v.
  Hint: For a triangle of three sides a, b, c with angles t between a and b, the formula to determine t is
  cos t = (a² + b² - c²)/(2 a b)
  
  Compute the sum s_v of angles φ .
- If, for all v,
  | s_v - 2 π | < 0.0001,
  stop and we have our r_i.
  (Here π = pi = 180 degrees.)
  If, for some v,
  s_v < 2 π - 0.0001,
  set new r_i = old r_i + 0.0001
  
  If, for some v,
  s_v > 2 π + 0.0001,
  set new r_i = old r_i - 0.0001
  Repeat Step i.
Layout the disc representation using r_v and R.