Algorithm for disc representation for a planar graph

An outline

• Input: a drawing of a planar graph in the plane
• Outline of an algorithm for computing the radius of discs associated with vertices
  
  1. Triangulate the planar drawing by adding edges if necessary. The resulting drawing has all faces being triangles.
  2. Assign weight 1/2 to each vertex, i.e.,
     set   r_i = 1/2   for each vertex v_i.
  3. Compute, for each interior vertex v_i, the sum s_i of angles at v_i in faces incident to v_i.
     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)
     Hint: For a vertex incident to the outerface, extra care should be given for incident angles.
  4. If, for all i,    
     | s_i - 2 π | < 0.0001,
     stop and we have our r_i.
     (Here π = pi = 180 degrees.)
     If, for some i,    
     s_i < 2 π - 0.0001,
     set     new r_i = old r_i - 0.0001
     and the three vertices, say, v₁, v₂, v₃, incident to the outerface, with
         new r₁ = new r₂ = new r₃ = old r₁ + 0.0001/3
     Repeat Step 2.
• Layout the disc representation using r_i.