


from dataclasses import dataclass
from typing import Tuple

import numpy as np
import torch


@dataclass
class 		_SCREAMING_SNAKE_CASE     :
     __SCREAMING_SNAKE_CASE     					=    42  # [batch_size x 3]
     __SCREAMING_SNAKE_CASE     					=    42  # [batch_size x 3]
     __SCREAMING_SNAKE_CASE     					=    42  # [batch_size x 3]
     __SCREAMING_SNAKE_CASE     					=    42  # [batch_size x 3]
     __SCREAMING_SNAKE_CASE     					=    42
     __SCREAMING_SNAKE_CASE     					=    42
     __SCREAMING_SNAKE_CASE     					=    42
     __SCREAMING_SNAKE_CASE     					=    42
     __SCREAMING_SNAKE_CASE     					=    42


     def        __snake_case(  self    ):
       assert self.x.shape[0] == self.y.shape[0] == self.z.shape[0] == self.origin.shape[0]
       assert self.x.shape[1] == self.y.shape[1] == self.z.shape[1] == self.origin.shape[1] == 3
       assert len(self.x.shape    ) == len(self.y.shape    ) == len(self.z.shape    ) == len(self.origin.shape    ) == 2


     def        __snake_case(  self    ):
       return torch.from_numpy(np.array([self.width, self.height]							,					dtype=np.floataa    )    )


     def        __snake_case(  self    ):
       return torch.from_numpy(np.array([self.x_fov, self.y_fov]							,					dtype=np.floataa    )    )


     def        __snake_case(  self    ):
       _UpperCAmelCase		:							str							=     torch.arange(self.height * self.width    )
       _UpperCAmelCase		:							Tuple							=     torch.stack(
           [
               pixel_indices % self.width,
               torch.div(A_							,					self.width							,					rounding_mode="""trunc"""    ),
           ]							,					axis=1							,					)
       return coords


     @property
     def        __snake_case(  self    ):
       _UpperCAmelCase,*_UpperCAmelCase		:							List[Any]							=     self.shape
       _UpperCAmelCase		:							Union[str, Any]							=     int(np.prod(A_    )    )

       _UpperCAmelCase		:							Optional[int]							=     self.get_image_coords()
       _UpperCAmelCase		:							Any							=     torch.broadcast_to(coords.unsqueeze(0    )							,					[batch_size * inner_batch_size, *coords.shape]    )
       _UpperCAmelCase		:							int							=     self.get_camera_rays(A_    )

       _UpperCAmelCase		:							Dict							=     rays.view(A_							,					inner_batch_size * self.height * self.width							,					2							,					3    )

       return rays


     def        __snake_case(  self							,					A_    ):
       _UpperCAmelCase,*_UpperCAmelCase,_UpperCAmelCase		:							Optional[int]							=     coords.shape
       assert n_coords == 2
       assert batch_size == self.origin.shape[0]

       _UpperCAmelCase		:							Tuple							=     coords.view(A_							,					-1							,					2    )

       _UpperCAmelCase		:							List[str]							=     self.resolution()
       _UpperCAmelCase		:							List[Any]							=     self.fov()

       _UpperCAmelCase		:							Tuple							=     (flat.float() / (res - 1)) * 2 - 1
       _UpperCAmelCase		:							Tuple							=     fracs * torch.tan(fov / 2    )

       _UpperCAmelCase		:							Optional[Any]							=     fracs.view(A_							,					-1							,					2    )
       _UpperCAmelCase		:							Any							=     (
           self.z.view(A_							,					1							,					3    )
           + self.x.view(A_							,					1							,					3    ) * fracs[:, :, :1]
           + self.y.view(A_							,					1							,					3    ) * fracs[:, :, 1:]
       )
       _UpperCAmelCase		:							Optional[int]							=     directions / directions.norm(dim=-1							,					keepdim=A_    )
       _UpperCAmelCase		:							str							=     torch.stack(
           [
               torch.broadcast_to(self.origin.view(A_							,					1							,					3    )							,					[batch_size, directions.shape[1], 3]    ),
               directions,
           ]							,					dim=2							,					)
       return rays.view(A_							,					*A_							,					2							,					3    )




     def        __snake_case(  self							,					A_							,					A_    ):
       assert width * self.height == height * self.width, "The aspect ratio should not change."
       return DifferentiableProjectiveCamera(
           origin=self.origin							,					x=self.x							,					y=self.y							,					z=self.z							,					width=A_							,					height=A_							,					x_fov=self.x_fov							,					y_fov=self.y_fov							,					)




def    a__    (		snake_case__					:							int     ):
  _UpperCAmelCase		:							List[str]							=     []
  _UpperCAmelCase		:							Union[str, Any]							=     []
  _UpperCAmelCase		:							int							=     []
  _UpperCAmelCase		:							List[str]							=     []
  for theta in np.linspace(0						,   2 * np.pi						,   num=20     ):
    _UpperCAmelCase		:							Any							=     np.array([np.sin(snake_case__     ), np.cos(snake_case__     ), -0.5]     )
    z /= np.sqrt(np.sum(z**2     )     )
    _UpperCAmelCase		:							Optional[Any]							=     -z * 4
    _UpperCAmelCase		:							Union[str, Any]							=     np.array([np.cos(snake_case__     ), -np.sin(snake_case__     ), 0.0]     )
    _UpperCAmelCase		:							Optional[Any]							=     np.cross(snake_case__						,   snake_case__     )
    origins.append(snake_case__     )
    xs.append(snake_case__     )
    ys.append(snake_case__     )
    zs.append(snake_case__     )
  return DifferentiableProjectiveCamera(
      origin=torch.from_numpy(np.stack(snake_case__						,   axis=0     )     ).float()						,   x=torch.from_numpy(np.stack(snake_case__						,   axis=0     )     ).float()						,   y=torch.from_numpy(np.stack(snake_case__						,   axis=0     )     ).float()						,   z=torch.from_numpy(np.stack(snake_case__						,   axis=0     )     ).float()						,   width=snake_case__						,   height=snake_case__						,   x_fov=0.7						,   y_fov=0.7						,   shape=(1, len(snake_case__     ))						,   )









def    a__    (		snake_case__					:							int						,   snake_case__					:							int     ):
  return x if y == 0 else greatest_common_divisor(snake_case__						,   x % y     )




def    a__    (		snake_case__					:							int						,   snake_case__					:							int     ):
  return (x * y) // greatest_common_divisor(snake_case__						,   snake_case__     )




def    a__    (		snake_case__					:							int = 20     ):
  _UpperCAmelCase		:							Union[str, Any]							=     1
  for i in range(1						,   n + 1     ):
    _UpperCAmelCase		:							Dict							=     lcm(snake_case__						,   snake_case__     )
  return g


if __name__ == "__main__":
 print(F'{solution() = }')





