code: def centroid_refine_triangulation_by_triangles(self, triangles):
          
  
          # Remove duplicates from the list of triangles
  
          triangles = np.unique(np.array(triangles))
  
          mlons, mlats = self.face_midpoints(simplices=self.simplices[triangles])
  
          lonv1 = np.concatenate((self.lons, mlons), axis=0)
          latv1 = np.concatenate((self.lats, mlats), axis=0)
  
          return lonv1, latv1
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].