content: import torch
  import torch.nn.functional as F
  import numpy as np
  
  
  def pad_camera_extrinsics_4x4(extrinsics):
      if extrinsics.shape[-2] == 4:
          return extrinsics
      padding = torch.tensor([[0, 0, 0, 1]]).to(extrinsics)
      if extrinsics.ndim == 3:
          padding = padding.unsqueeze(0).repeat(extrinsics.shape[0], 1, 1)
      extrinsics = torch.cat([extrinsics, padding], dim=-2)
      return extrinsics
  
  
  def center_looking_at_camera_pose(camera_position: torch.Tensor, look_at: torch.Tensor = None, up_world: torch.Tensor = None):
      """
      Create OpenGL camera extrinsics from camera locations and look-at position.
  
      camera_position: (M, 3) or (3,)
      look_at: (3)
      up_world: (3)
      return: (M, 3, 4) or (3, 4)
      """
      # by default, looking at the origin and world up is z-axis
      if look_at is None:
          look_at = torch.tensor([0, 0, 0], dtype=torch.float32)
      if up_world is None:
          up_world = torch.tensor([0, 0, 1], dtype=torch.float32)
      if camera_position.ndim == 2:
          look_at = look_at.unsqueeze(0).repeat(camera_position.shape[0], 1)
          up_world = up_world.unsqueeze(0).repeat(camera_position.shape[0], 1)
  
      # OpenGL camera: z-backward, x-right, y-up
      z_axis = camera_position - look_at
      z_axis = F.normalize(z_axis, dim=-1).float()
      x_axis = torch.linalg.cross(up_world, z_axis, dim=-1)
      x_axis = F.normalize(x_axis, dim=-1).float()
      y_axis = torch.linalg.cross(z_axis, x_axis, dim=-1)
      y_axis = F.normalize(y_axis, dim=-1).float()
  
      extrinsics = torch.stack([x_axis, y_axis, z_axis, camera_position], dim=-1)
      extrinsics = pad_camera_extrinsics_4x4(extrinsics)
      return extrinsics
  
  
  def spherical_camera_pose(azimuths: np.ndarray, elevations: np.ndarray, radius=2.5):
      azimuths = np.deg2rad(azimuths)
      elevations = np.deg2rad(elevations)
  
      xs = radius * np.cos(elevations) * np.cos(azimuths)
      ys = radius * np.cos(elevations) * np.sin(azimuths)
      zs = radius * np.sin(elevations)
  
      cam_locations = np.stack([xs, ys, zs], axis=-1)
      cam_locations = torch.from_numpy(cam_locations).float()
  
      c2ws = center_looking_at_camera_pose(cam_locations)
      return c2ws
  
  
  def get_circular_camera_poses(M=120, radius=2.5, elevation=30.0):
      # M: number of circular views
      # radius: camera dist to center
      # elevation: elevation degrees of the camera
      # return: (M, 4, 4)
      assert M > 0 and radius > 0
  
      elevation = np.deg2rad(elevation)
  
      camera_positions = []
      for i in range(M):
          azimuth = 2 * np.pi * i / M
          x = radius * np.cos(elevation) * np.cos(azimuth)
          y = radius * np.cos(elevation) * np.sin(azimuth)
          z = radius * np.sin(elevation)
          camera_positions.append([x, y, z])
      camera_positions = np.array(camera_positions)
      camera_positions = torch.from_numpy(camera_positions).float()
      extrinsics = center_looking_at_camera_pose(camera_positions)
      return extrinsics
  
  def FOV_to_intrinsics(fov, device='cpu'):
      """
      Creates a 3x3 camera intrinsics matrix from the camera field of view, specified in degrees.
      Note the intrinsics are returned as normalized by image size, rather than in pixel units.
      Assumes principal point is at image center.
      """
      focal_length = 0.5 / np.tan(np.deg2rad(fov) * 0.5)
      intrinsics = torch.tensor([[focal_length, 0, 0.5], [0, focal_length, 0.5], [0, 0, 1]], device=device)
      return intrinsics
      
  def oribt_camera_poses_to_input_cameras(azimuths, elevations, batch_size=1, radius=4.0, fov=30.0):
      """
          Args:
              azimuths (list)
              elevations (list)
              radius (float or list)
      """
      # (0, 180/-180, 0) -> (0, 180, 360)
      azimuths = [360 + angle if angle < 0 else angle for angle in azimuths]
      azimuths = np.array(azimuths).astype(float)
      elevations = -np.array(elevations).astype(float)
      if isinstance(radius, list):
          radius = np.array(radius).astype(float)
  
      c2ws = spherical_camera_pose(azimuths, elevations, radius)
      c2ws = c2ws.float().flatten(-2)
  
      Ks = FOV_to_intrinsics(fov).unsqueeze(0).repeat(azimuths.shape[0], 1, 1).float().flatten(-2)
  
      extrinsics = c2ws[:, :12]
      intrinsics = torch.stack([Ks[:, 0], Ks[:, 4], Ks[:, 2], Ks[:, 5]], dim=-1)
      cameras = torch.cat([extrinsics, intrinsics], dim=-1)
  
      return cameras.unsqueeze(0).repeat(batch_size, 1, 1)
  