






"""simple docstring"""


from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
_UpperCamelCase :    Dict      =			20

# Time between time steps in seconds
_UpperCamelCase :    Optional[int]      =			INTERVAL / 10_00






class __a :



  def __init__(self	:		Optional[int]					,      lowercase_	:		float					,      lowercase_	:		float					,      lowercase_	:		float					,      lowercase_	:		float					,      lowercase_	:		float = 1.0					,      lowercase_	:		float = 1.0					,      lowercase_	:		str = "blue"					,      )		->				None:
   __A						:							List[str]      =						position_x
   __A						:							Tuple      =						position_y
   __A						:							List[Any]      =						velocity_x
   __A						:							Union[str, Any]      =						velocity_y
   __A						:							Dict      =						mass
   __A						:							List[str]      =						size
   __A						:							Union[str, Any]      =						color



  @property
  def     A_					(self	:		Optional[Any] )		->				tuple[float, float]:
   return self.position_x, self.position_y



  @property
  def     A_					(self	:		Optional[int] )		->				tuple[float, float]:
   return self.velocity_x, self.velocity_y



  def     A_					(self	:		Optional[int]					,      lowercase_	:		float					,      lowercase_	:		float					,      lowercase_	:		float )		->				None:
   self.velocity_x += force_x * delta_time
   self.velocity_y += force_y * delta_time



  def     A_					(self	:		int					,      lowercase_	:		float )		->				None:
   self.position_x += self.velocity_x * delta_time
   self.position_y += self.velocity_y * delta_time






class __a :



  def __init__(self	:		Any					,      lowercase_	:		list[Body]					,      lowercase_	:		float = 1.0					,      lowercase_	:		float = 1.0					,      lowercase_	:		float = 0.0					,      )		->				None:
   __A						:							int      =						bodies
   __A						:							Tuple      =						gravitation_constant
   __A						:							List[str]      =						time_factor
   __A						:							Optional[Any]      =						softening_factor



  def __len__(self	:		Dict )		->				int:
   return len(self.bodies )



  def     A_					(self	:		str					,      lowercase_	:		float )		->				None:
   for bodya in self.bodies:
    __A						:							Optional[Any]      =						0.0
    __A						:							List[Any]      =						0.0
    for bodya in self.bodies:
     if bodya != bodya:
      __A						:							str      =						bodya.position_x - bodya.position_x
      __A						:							str      =						bodya.position_y - bodya.position_y

      # Calculation of the distance using Pythagoras's theorem
      # Extra factor due to the softening technique
      __A						:							List[str]      =						(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(lowercase_					,      lowercase_					,      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 					a			(  UpperCamelCase_				:					BodySystem  ,						UpperCamelCase_				:					float  ,						UpperCamelCase_				:					list[plt.Circle]		)	->		None:
 body_system.update_system(UpperCamelCase_		)

 # Update the positions of the patches
 for patch, body in zip(UpperCamelCase_  ,						body_system.bodies		):
  __A						:							Optional[Any]      =						(body.position_x, body.position_y)




def 					a			(  UpperCamelCase_				:					str  ,						UpperCamelCase_				:					BodySystem  ,						UpperCamelCase_				:					float = -1  ,						UpperCamelCase_				:					float = 1  ,						UpperCamelCase_				:					float = -1  ,						UpperCamelCase_				:					float = 1  ,						)	->		None:
 __A						:							Union[str, Any]      =						plt.figure()
 fig.canvas.set_window_title(UpperCamelCase_		)
 __A						:							List[Any]      =						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
 __A						:							List[str]      =						[
     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(UpperCamelCase_		)

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

 __A						:							int      =						animation.FuncAnimation(  # noqa: F841
     UpperCamelCase_  ,						UpperCamelCase_  ,						interval=UpperCamelCase_  ,						blit=UpperCamelCase_		)

 plt.show()




def 					a			(  )	->		BodySystem:
 __A						:							Optional[int]      =						0.9700436
 __A						:							str      =						-0.24308753
 __A						:							List[Any]      =						0.466203685
 __A						:							List[str]      =						0.43236573

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




def 					a			(  )	->		BodySystem:
 __A						:							Dict      =						7.3_476E22
 __A						:							List[Any]      =						5.972E24
 __A						:							str      =						1_0_2_2
 __A						:							Dict      =						3_8_4_3_9_9_0_0_0
 __A						:							Dict      =						6.674E-11

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

 __A						:							Tuple      =						Body(-earth_moon_distance  ,						0  ,						0  ,						UpperCamelCase_  ,						UpperCamelCase_  ,						1_0_0_0_0_0_0_0  ,						'grey'		)
 __A						:							str      =						Body(0  ,						0  ,						0  ,						UpperCamelCase_  ,						UpperCamelCase_  ,						5_0_0_0_0_0_0_0  ,						'blue'		)
 return BodySystem([earth, moon]  ,						UpperCamelCase_  ,						time_factor=1_0_0_0_0_0_0		)




def 					a			(  )	->		BodySystem:
 __A						:							Union[str, Any]      =						[]
 for _ in range(1_0		):
  __A						:							Dict      =						random.uniform(-0.5  ,						0.5		)
  __A						:							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		)  ,						UpperCamelCase_  ,						UpperCamelCase_  ,						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(UpperCamelCase_  ,						0.01  ,						1_0  ,						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)













"""simple docstring"""


import numpy as np

_UpperCamelCase :    int      =			[
    ["""a""", """b""", """c""", """d""", """e"""],
    ["""f""", """g""", """h""", """i""", """k"""],
    ["""l""", """m""", """n""", """o""", """p"""],
    ["""q""", """r""", """s""", """t""", """u"""],
    ["""v""", """w""", """x""", """y""", """z"""],
]






class __a :



  def __init__(self	:		Any )		->				None:
   __A						:							Any      =						np.array(lowercase_ )



  def     A_					(self	:		Dict					,      lowercase_	:		str )		->				np.ndarray:
   __A     ,       __A						:							Any      =						np.where(letter == self.SQUARE )
   __A						:							Tuple      =						np.concatenate([indexa + 1, indexa + 1] )
   return indexes



  def     A_					(self	:		Tuple					,      lowercase_	:		int					,      lowercase_	:		int )		->				str:
   return self.SQUARE[indexa - 1, indexa - 1]



  def     A_					(self	:		Union[str, Any]					,      lowercase_	:		str )		->				str:
   __A						:							Any      =						message.lower()
   __A						:							Optional[Any]      =						message.replace('j'					,      'i' )

   __A						:							Union[str, Any]      =						''
   for letter_index in range(len(lowercase_ ) ):
    if message[letter_index] != " ":
     __A						:							Union[str, Any]      =						self.letter_to_numbers(message[letter_index] )
     __A						:							Union[str, Any]      =						encoded_message + str(numbers[0] ) + str(numbers[1] )
    elif message[letter_index] == " ":
     __A						:							Optional[Any]      =						encoded_message + ' '

   return encoded_message



  def     A_					(self	:		Dict					,      lowercase_	:		str )		->				str:
   __A						:							List[str]      =						message.replace(' '					,      '  ' )
   __A						:							List[Any]      =						''
   for numbers_index in range(int(len(lowercase_ ) / 2 ) ):
    if message[numbers_index * 2] != " ":
     __A						:							Any      =						message[numbers_index * 2]
     __A						:							Tuple      =						message[numbers_index * 2 + 1]

     __A						:							Dict      =						self.numbers_to_letter(int(lowercase_ )					,      int(lowercase_ ) )
     __A						:							Union[str, Any]      =						decoded_message + letter
    elif message[numbers_index * 2] == " ":
     __A						:							int      =						decoded_message + ' '

   return decoded_message





