code: def mesh_laplacian_smoothing(meshes, method: str = "uniform"):
      r"""
      Computes the laplacian smoothing objective for a batch of meshes.
      This function supports three variants of Laplacian smoothing,
      namely with uniform weights("uniform"), with cotangent weights ("cot"),
      and cotangent curvature ("cotcurv").For more details read [1, 2].
  
      Args:
          meshes: Meshes object with a batch of meshes.
          method: str specifying the method for the laplacian.
      Returns:
          loss: Average laplacian smoothing loss across the batch.
          Returns 0 if meshes contains no meshes or all empty meshes.
  
      Consider a mesh M = (V, F), with verts of shape Nx3 and faces of shape Mx3.
      The Laplacian matrix L is a NxN tensor such that LV gives a tensor of vectors:
      for a uniform Laplacian, LuV[i] points to the centroid of its neighboring
      vertices, a cotangent Laplacian LcV[i] is known to be an approximation of
      the surface normal, while the curvature variant LckV[i] scales the normals
      by the discrete mean curvature. For vertex i, assume S[i] is the set of
      neighboring vertices to i, a_ij and b_ij are the "outside" angles in the
      two triangles connecting vertex v_i and its neighboring vertex v_j
      for j in S[i], as seen in the diagram below.
  
      .. code-block:: python
  
                 a_ij
                  /\
                 /  \
                /    \
               /      \
          v_i /________\ v_j
              \        /
               \      /
                \    /
                 \  /
                  \/
                 b_ij
  
          The definition of the Laplacian is LV[i] = sum_j w_ij (v_j - v_i)
          For the uniform variant,    w_ij = 1 / |S[i]|
          For the cotangent variant,
              w_ij = (cot a_ij + cot b_ij) / (sum_k cot a_ik + cot b_ik)
          For the cotangent curvature, w_ij = (cot a_ij + cot b_ij) / (4 A[i])
          where A[i] is the sum of the areas of all triangles containing vertex v_i.
  
      There is a nice trigonometry identity to compute cotangents. Consider a triangle
      with side lengths A, B, C and angles a, b, c.
  
      .. code-block:: python
  
                 c
                /|\
               / | \
              /  |  \
           B /  H|   \ A
            /    |    \
           /     |     \
          /a_____|_____b\
                 C
  
          Then cot a = (B^2 + C^2 - A^2) / 4 * area
          We know that area = CH/2, and by the law of cosines we have
  
          A^2 = B^2 + C^2 - 2BC cos a => B^2 + C^2 - A^2 = 2BC cos a
  
          Putting these together, we get:
  
          B^2 + C^2 - A^2     2BC cos a
          _______________  =  _________ = (B/H) cos a = cos a / sin a = cot a
             4 * area            2CH
  
  
      [1] Desbrun et al, "Implicit fairing of irregular meshes using diffusion
      and curvature flow", SIGGRAPH 1999.
  
      [2] Nealan et al, "Laplacian Mesh Optimization", Graphite 2006.
      """
  
      if meshes.isempty():
          return torch.tensor(
              [0.0],
              dtype=torch.float32,
              device=meshes.device,
              requires_grad=True,
          )
  
      N = len(meshes)
      verts_packed = meshes.verts_packed()  # (sum(V_n), 3)
      faces_packed = meshes.faces_packed()  # (sum(F_n), 3)
      num_verts_per_mesh = meshes.num_verts_per_mesh()  # (N,)
      verts_packed_idx = (
          meshes.verts_packed_to_mesh_idx()
      )  # (sum(V_n),)
      weights = num_verts_per_mesh.gather(
          0, verts_packed_idx
      )  # (sum(V_n),)
      weights = 1.0 / weights.float()
  
      # We don't want to backprop through the computation of the Laplacian;
      # just treat it as a magic constant matrix that is used to transform
      # verts into normals
      with torch.no_grad():
          if method == "uniform":
              L = meshes.laplacian_packed()
          elif method in ["cot", "cotcurv"]:
              L, inv_areas = cot_laplacian(verts_packed, faces_packed)
              if method == "cot":
                  norm_w = (
                      torch.sparse.sum(L, dim=1).to_dense().view(-1, 1)
                  )
                  idx = norm_w > 0
                  # pyre-fixme[58]: `/` is not supported for operand types `float` and
                  #  `Tensor`.
                  norm_w[idx] = 1.0 / norm_w[idx]
              else:
                  L_sum = (
                      torch.sparse.sum(L, dim=1).to_dense().view(-1, 1)
                  )
                  norm_w = 0.25 * inv_areas
          else:
              raise ValueError(
                  "Method should be one of {uniform, cot, cotcurv}"
              )
  
      if method == "uniform":
          loss = L.mm(verts_packed)
      elif method == "cot":
          # pyre-fixme[61]: `norm_w` is undefined, or not always defined.
          loss = L.mm(verts_packed) * norm_w - verts_packed
      elif method == "cotcurv":
          # pyre-fixme[61]: `norm_w` may not be initialized here.
          loss = (L.mm(verts_packed) - L_sum * verts_packed) * norm_w
      loss = loss.norm(dim=1)
  
      loss = loss * weights
      return loss.sum() / N
  
label_desc: Use image node_id 1 for calling a global function with example usage: mesh_laplacian_smoothing(meshes, method) and returns: unknown, torch