def midpoint_refine_triangulation_by_vertices(self, vertices):
        """
        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
        """

        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
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
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