from __future__ import annotations





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase    )->	str:






      '''simple docstring'''





      UpperCAmelCase	,    UpperCAmelCase	,    UpperCAmelCase     :			List[str]     	='''''', '''''', ''''''
      UpperCAmelCase     :			Union[str, Any]     	=[]

      for character in message_part:
            tmp.append(character_to_number[character]    )

      for each in tmp:
            one += each[0]
            two += each[1]
            three += each[2]

      return one + two + three





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase    )->	tuple[str, str, str]:






      '''simple docstring'''





      UpperCAmelCase	,    UpperCAmelCase     :			Optional[Any]     	='''''', ''''''
      UpperCAmelCase     :			Dict     	=[]

      for character in message_part:
            this_part += character_to_number[character]

      for digit in this_part:
            tmp += digit
            if len(__lowerCAmelCase    ) == len(__lowerCAmelCase    ):
                  result.append(__lowerCAmelCase    )
                  UpperCAmelCase     :			str     	=''''''

      return result[0], result[1], result[2]





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase    )->	tuple[str, str, dict[str, str], dict[str, str]]:






      '''simple docstring'''





      UpperCAmelCase     :			Any     	=alphabet.replace(''' '''		,       ''''''    ).upper()
      UpperCAmelCase     :			Dict     	=message.replace(''' '''		,       ''''''    ).upper()

      # Check length and characters
      if len(__lowerCAmelCase    ) != 27:
            raise KeyError('''Length of alphabet has to be 27.'''    )
      for each in message:
            if each not in alphabet:
                  raise ValueError('''Each message character has to be included in alphabet!'''    )

    # Generate dictionares
      UpperCAmelCase     :			int     	=(
          '''111''',
          '''112''',
          '''113''',
          '''121''',
          '''122''',
          '''123''',
          '''131''',
          '''132''',
          '''133''',
          '''211''',
          '''212''',
          '''213''',
          '''221''',
          '''222''',
          '''223''',
          '''231''',
          '''232''',
          '''233''',
          '''311''',
          '''312''',
          '''313''',
          '''321''',
          '''322''',
          '''323''',
          '''331''',
          '''332''',
          '''333''',
      )
      UpperCAmelCase     :			List[Any]     	={}
      UpperCAmelCase     :			str     	={}
      for letter, number in zip(__lowerCAmelCase		,       __lowerCAmelCase    ):
            UpperCAmelCase     :			int     	=number
            UpperCAmelCase     :			Union[str, Any]     	=letter

      return message, alphabet, character_to_number, number_to_character





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase = "ABCDEFGHIJKLMNOPQRSTUVWXYZ."		,       __lowerCAmelCase = 5    )->	str:






      '''simple docstring'''





      UpperCAmelCase	,    UpperCAmelCase	,    UpperCAmelCase	,    UpperCAmelCase     :			Dict     	=__prepare(
          __lowerCAmelCase		,       __lowerCAmelCase    )
      UpperCAmelCase	,    UpperCAmelCase     :			List[Any]     	='''''', ''''''

      for i in range(0		,       len(__lowerCAmelCase    ) + 1		,       __lowerCAmelCase    ):
            encrypted_numeric += __encrypt_part(
                message[i : i + period]		,       __lowerCAmelCase    )

      for i in range(0		,       len(__lowerCAmelCase    )		,       3    ):
            encrypted += number_to_character[encrypted_numeric[i : i + 3]]

      return encrypted





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase = "ABCDEFGHIJKLMNOPQRSTUVWXYZ."		,       __lowerCAmelCase = 5    )->	str:






      '''simple docstring'''





      UpperCAmelCase	,    UpperCAmelCase	,    UpperCAmelCase	,    UpperCAmelCase     :			List[Any]     	=__prepare(
          __lowerCAmelCase		,       __lowerCAmelCase    )
      UpperCAmelCase     :			Dict     	=[]
      UpperCAmelCase     :			Optional[Any]     	=''''''

      for i in range(0		,       len(__lowerCAmelCase    ) + 1		,       __lowerCAmelCase    ):
            UpperCAmelCase	,    UpperCAmelCase	,    UpperCAmelCase     :			Tuple     	=__decrypt_part(message[i : i + period]		,       __lowerCAmelCase    )

            for j in range(0		,       len(__lowerCAmelCase    )    ):
                  decrypted_numeric.append(a[j] + b[j] + c[j]    )

      for each in decrypted_numeric:
            decrypted += number_to_character[each]

      return decrypted


if __name__ == "__main__":
     __snake_case         =     '''DEFEND THE EAST WALL OF THE CASTLE.'''
     __snake_case         =     encrypt_message(msg, '''EPSDUCVWYM.ZLKXNBTFGORIJHAQ''')
     __snake_case         =     decrypt_message(encrypted, '''EPSDUCVWYM.ZLKXNBTFGORIJHAQ''')
     print(f'Encrypted: {encrypted}\nDecrypted: {decrypted}')

from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
__snake_case         =     20

# Time between time steps in seconds
__snake_case         =     INTERVAL / 10_00







class       __snake_case :
 def __init__(	self		, snake_case__		, snake_case__		, snake_case__		, snake_case__		, snake_case__ = 1.0		, snake_case__ = 1.0		, snake_case__ = "blue"		, )						->				None:
       '''simple docstring'''





       UpperCAmelCase     :			Any     	=position_x
       UpperCAmelCase     :			Dict     	=position_y
       UpperCAmelCase     :			List[Any]     	=velocity_x
       UpperCAmelCase     :			Optional[int]     	=velocity_y
       UpperCAmelCase     :			List[str]     	=mass
       UpperCAmelCase     :			Any     	=size
       UpperCAmelCase     :			List[str]     	=color
 @property
 def 						UpperCAmelCase__							(	self    )						->				tuple[float, float]:
       '''simple docstring'''





       return self.position_x, self.position_y
 @property
 def 						UpperCAmelCase__							(	self    )						->				tuple[float, float]:
       '''simple docstring'''





       return self.velocity_x, self.velocity_y
 def 						UpperCAmelCase__							(	self		, snake_case__		, snake_case__		, snake_case__    )						->				None:
       '''simple docstring'''





       self.velocity_x += force_x * delta_time
       self.velocity_y += force_y * delta_time






 def 						UpperCAmelCase__							(	self		, snake_case__    )						->				None:
       '''simple docstring'''





       self.position_x += self.velocity_x * delta_time
       self.position_y += self.velocity_y * delta_time







class       __snake_case :
 def __init__(	self		, snake_case__		, snake_case__ = 1.0		, snake_case__ = 1.0		, snake_case__ = 0.0		, )						->				None:
       '''simple docstring'''





       UpperCAmelCase     :			Tuple     	=bodies
       UpperCAmelCase     :			List[str]     	=gravitation_constant
       UpperCAmelCase     :			int     	=time_factor
       UpperCAmelCase     :			List[Any]     	=softening_factor
 def __len__(	self    )						->				int:
       '''simple docstring'''





       return len(self.bodies    )






 def 						UpperCAmelCase__							(	self		, snake_case__    )						->				None:
       '''simple docstring'''





       for bodya in self.bodies:
             UpperCAmelCase     :			Tuple     	=0.0
             UpperCAmelCase     :			Union[str, Any]     	=0.0
             for bodya in self.bodies:
                   if bodya != bodya:
                         UpperCAmelCase     :			Optional[Any]     	=bodya.position_x - bodya.position_x
                         UpperCAmelCase     :			Any     	=bodya.position_y - bodya.position_y

                         # Calculation of the distance using Pythagoras's theorem
                         # Extra factor due to the softening technique
                         UpperCAmelCase     :			Optional[Any]     	=(dif_x**2 + dif_y**2 + self.softening_factor) ** (
                             1 / 2
                         )

                         # Newton's law of universal gravitation.
                         force_x += (
                             self.gravitation_constant * bodya.mass * dif_x / distance**3
                         )
                         force_y += (
                             self.gravitation_constant * bodya.mass * dif_y / distance**3
                         )

            # Update the body's velocity once all the force components have been added
             bodya.update_velocity(snake_case__		, snake_case__		, delta_time * self.time_factor    )

       # Update the positions only after all the velocities have been updated
       for body in self.bodies:
             body.update_position(delta_time * self.time_factor    )





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase    )->	None:






      '''simple docstring'''





      body_system.update_system(__lowerCAmelCase    )

      # Update the positions of the patches
      for patch, body in zip(__lowerCAmelCase		,       body_system.bodies    ):
            UpperCAmelCase     :			Dict     	=(body.position_x, body.position_y)





def 	lowerCAmelCase_			(					__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase = -1		,       __lowerCAmelCase = 1		,       __lowerCAmelCase = -1		,       __lowerCAmelCase = 1		,       )->	None:






      '''simple docstring'''





      UpperCAmelCase     :			Optional[Any]     	=plt.figure()
      fig.canvas.set_window_title(__lowerCAmelCase    )
      UpperCAmelCase     :			Dict     	=plt.axes(
          xlim=(x_start, x_end)		,       ylim=(y_start, y_end)    )  # Set section to be plotted
      plt.gca().set_aspect('''equal'''    )  # Fix aspect ratio

      # Each body is drawn as a patch by the plt-function
      UpperCAmelCase     :			Tuple     	=[
          plt.Circle((body.position_x, body.position_y)		,       body.size		,       fc=body.color    )
          for body in body_system.bodies
      ]

      for patch in patches:
            ax.add_patch(__lowerCAmelCase    )

      # Function called at each step of the animation
      def update(__lowerCAmelCase    ) -> list[plt.Circle]:
            update_step(__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase    )
            return patches

      UpperCAmelCase     :			List[Any]     	=animation.FuncAnimation(  # noqa: F841
          __lowerCAmelCase		,       __lowerCAmelCase		,       interval=__lowerCAmelCase		,       blit=__lowerCAmelCase    )

      plt.show()





def 	lowerCAmelCase_			(					)->	BodySystem:






      '''simple docstring'''





      UpperCAmelCase     :			int     	=0.9700436
      UpperCAmelCase     :			Dict     	=-0.24308753
      UpperCAmelCase     :			List[Any]     	=0.466203685
      UpperCAmelCase     :			Dict     	=0.43236573

      UpperCAmelCase     :			Optional[Any]     	=[
          Body(__lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase		,       __lowerCAmelCase		,       size=0.2		,       color='''red'''    ),
          Body(-position_x		,       -position_y		,       __lowerCAmelCase		,       __lowerCAmelCase		,       size=0.2		,       color='''green'''    ),
          Body(0		,       0		,       -2 * velocity_x		,       -2 * velocity_y		,       size=0.2		,       color='''blue'''    ),
      ]
      return BodySystem(__lowerCAmelCase		,       time_factor=3    )





def 	lowerCAmelCase_			(					)->	BodySystem:






      '''simple docstring'''





      UpperCAmelCase     :			List[str]     	=7.3_4_7_6e2_2
      UpperCAmelCase     :			Tuple     	=5.9_7_2e2_4
      UpperCAmelCase     :			str     	=10_22
      UpperCAmelCase     :			Any     	=3_84_39_90_00
      UpperCAmelCase     :			List[Any]     	=6.6_7_4e-1_1

      # Calculation of the respective velocities so that total impulse is zero,
      # i.e. the two bodies together don't move
      UpperCAmelCase     :			Optional[int]     	=earth_mass * velocity_dif / (earth_mass + moon_mass)
      UpperCAmelCase     :			Optional[Any]     	=moon_velocity - velocity_dif

      UpperCAmelCase     :			Optional[int]     	=Body(-earth_moon_distance		,       0		,       0		,       __lowerCAmelCase		,       __lowerCAmelCase		,       10_00_00_00		,       '''grey'''    )
      UpperCAmelCase     :			Dict     	=Body(0		,       0		,       0		,       __lowerCAmelCase		,       __lowerCAmelCase		,       50_00_00_00		,       '''blue'''    )
      return BodySystem([earth, moon]		,       __lowerCAmelCase		,       time_factor=1_00_00_00    )





def 	lowerCAmelCase_			(					)->	BodySystem:






      '''simple docstring'''





      UpperCAmelCase     :			Optional[int]     	=[]
      for _ in range(10    ):
            UpperCAmelCase     :			Union[str, Any]     	=random.uniform(-0.5		,       0.5    )
            UpperCAmelCase     :			List[Any]     	=random.uniform(-0.5		,       0.5    )

            # Bodies are created pairwise with opposite velocities so that the
            # total impulse remains zero
            bodies.append(
                Body(
                    random.uniform(-0.5		,       0.5    )		,       random.uniform(-0.5		,       0.5    )		,       __lowerCAmelCase		,       __lowerCAmelCase		,       size=0.05		,       )    )
            bodies.append(
                Body(
                    random.uniform(-0.5		,       0.5    )		,       random.uniform(-0.5		,       0.5    )		,       -velocity_x		,       -velocity_y		,       size=0.05		,       )    )
      return BodySystem(__lowerCAmelCase		,       0.01		,       10		,       0.1    )


if __name__ == "__main__":
     plot('''Figure-8 solution to the 3-body-problem''', example_a(), -2, 2, -2, 2)
     plot(
         '''Moon\'s orbit around the earth''',
         example_a(),
         -4_30_00_00_00,
         4_30_00_00_00,
         -4_30_00_00_00,
         4_30_00_00_00,
     )
     plot('''Random system with many bodies''', example_a(), -1.5, 1.5, -1.5, 1.5)
