code: def midpoint_refine_triangulation_by_vertices(self, vertices):
          
  
          xi, yi = self.segment_midpoints_by_vertices(vertices=vertices)
  
          x_v1 = np.concatenate((self.x, xi), axis=0)
          y_v1 = np.concatenate((self.y, yi), axis=0)
  
          return x_v1, y_v1
docs: return points defining a refined triangulation obtained by bisection of all edges
          in the triangulation connected to any of the vertices in the list provided