LOCAL
    x_axis : direction;
    v      : direction;
    z      : direction;
    x_vec  : vector;
  END_LOCAL;
  
  IF  (NOT EXISTS(z_axis)) THEN
    RETURN  (?) ;
  ELSE
    z := normalise(z_axis);
    IF  NOT EXISTS(arg) THEN
      IF  ((z.direction_ratios <> [1.0,0.0,0.0]) AND
          (z.direction_ratios <> [-1.0,0.0,0.0]))  THEN
        v :=  dummy_gri || direction([1.0,0.0,0.0]);
      ELSE
        v := dummy_gri || direction([0.0,1.0,0.0]);
      END_IF;
    ELSE
      IF   (arg.dim <> 3) THEN
        RETURN  (?) ;
      END_IF;
      IF ((cross_product(arg,z).magnitude) = 0.0) THEN
        RETURN (?);
      ELSE
        v := normalise(arg);
      END_IF;
    END_IF;
    x_vec := scalar_times_vector(dot_product(v, z), z);
    x_axis := vector_difference(v, x_vec).orientation;
    x_axis := normalise(x_axis);
  END_IF;
  RETURN(x_axis);