code: def centroid_refine_triangulation_by_triangles(self, triangles):
          
  
          # Remove duplicates from the list of triangles
  
          triangles = np.unique(np.array(triangles))
  
          xi, yi = self.face_midpoints(simplices=self.simplices[triangles])
  
          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 that are associated with the triangles in the list provided.
  
          Notes
          -----
           The triangles are here represented as a single index.
           The vertices of triangle i are given by self.simplices[i].