

from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
lowercase     :       Optional[int]        =       20

# Time between time steps in seconds
lowercase     :       Dict        =       INTERVAL / 1000
class 		SCREAMING_SNAKE_CASE__					:







						"""simple docstring"""




						def __init__(				self , __UpperCamelCase , __UpperCamelCase , __UpperCamelCase , __UpperCamelCase , __UpperCamelCase = 1.0 , __UpperCamelCase = 1.0 , __UpperCamelCase = "blue" , )							->	None:




									'''simple docstring'''


									__UpperCamelCase    :    List[str]								=      position_x
									__UpperCamelCase    :    str								=      position_y
									__UpperCamelCase    :    Dict								=      velocity_x
									__UpperCamelCase    :    List[Any]								=      velocity_y
									__UpperCamelCase    :    Optional[int]								=      mass
									__UpperCamelCase    :    Union[str, Any]								=      size
									__UpperCamelCase    :    List[str]								=      color

						@property
						def __lowerCamelCase					(				self    )							->	tuple[float, float]:




									'''simple docstring'''


									return self.position_x, self.position_y

						@property
						def __lowerCamelCase					(				self    )							->	tuple[float, float]:




									'''simple docstring'''


									return self.velocity_x, self.velocity_y

						def __lowerCamelCase					(				self , __UpperCamelCase , __UpperCamelCase , __UpperCamelCase    )							->	None:




									'''simple docstring'''


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




						def __lowerCamelCase					(				self , __UpperCamelCase    )							->	None:




									'''simple docstring'''


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







class 		SCREAMING_SNAKE_CASE__					:







						"""simple docstring"""




						def __init__(				self , __UpperCamelCase , __UpperCamelCase = 1.0 , __UpperCamelCase = 1.0 , __UpperCamelCase = 0.0 , )							->	None:




									'''simple docstring'''


									__UpperCamelCase    :    Optional[Any]								=      bodies
									__UpperCamelCase    :    Dict								=      gravitation_constant
									__UpperCamelCase    :    Tuple								=      time_factor
									__UpperCamelCase    :    Dict								=      softening_factor

						def __len__(				self    )							->	int:




									'''simple docstring'''


									return len(self.bodies    )




						def __lowerCamelCase					(				self , __UpperCamelCase    )							->	None:




									'''simple docstring'''


									for bodya in self.bodies:
												__UpperCamelCase    :    int								=      0.0
												__UpperCamelCase    :    Dict								=      0.0
												for bodya in self.bodies:
															if bodya != bodya:
																		__UpperCamelCase    :    Union[str, 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(__UpperCamelCase , __UpperCamelCase , 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  UpperCAmelCase_   (_lowerCAmelCase :		BodySystem     ,			_lowerCAmelCase :		float     ,			_lowerCAmelCase :		list[plt.Circle]		):
			body_system.update_system(_lowerCAmelCase		)

			# Update the positions of the patches
			for patch, body in zip(_lowerCAmelCase     ,			body_system.bodies		):
						__UpperCamelCase    :    str								=      (body.position_x, body.position_y)







def  UpperCAmelCase_   (_lowerCAmelCase :		str     ,			_lowerCAmelCase :		BodySystem     ,			_lowerCAmelCase :		float = -1     ,			_lowerCAmelCase :		float = 1     ,			_lowerCAmelCase :		float = -1     ,			_lowerCAmelCase :		float = 1     ,			):
			__UpperCamelCase    :    Tuple								=      plt.figure()
			fig.canvas.set_window_title(_lowerCAmelCase		)
			__UpperCamelCase    :    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
			__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 :		int		) -> list[plt.Circle]:
						update_step(_lowerCAmelCase     ,			_lowerCAmelCase     ,			_lowerCAmelCase		)
						return patches

			__UpperCamelCase    :    Tuple								=      animation.FuncAnimation(  # noqa: F841
			    _lowerCAmelCase     ,			_lowerCAmelCase     ,			interval=_lowerCAmelCase     ,			blit=_lowerCAmelCase		)

			plt.show()







def  UpperCAmelCase_   ():
			__UpperCamelCase    :    List[str]								=      0.9_700_436
			__UpperCamelCase    :    Optional[int]								=      -0.24_308_753
			__UpperCamelCase    :    Union[str, Any]								=      0.466_203_685
			__UpperCamelCase    :    str								=      0.43_236_573

			__UpperCamelCase    :    List[str]								=      [
			    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  UpperCAmelCase_   ():
			__UpperCamelCase    :    Optional[Any]								=      7.34_76E22
			__UpperCamelCase    :    Dict								=      5.9_72E24
			__UpperCamelCase    :    List[Any]								=      10_22
			__UpperCamelCase    :    Tuple								=      3_84_39_90_00
			__UpperCamelCase    :    int								=      6.6_74E-11

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

			__UpperCamelCase    :    int								=      Body(-earth_moon_distance     ,			0     ,			0     ,			_lowerCAmelCase     ,			_lowerCAmelCase     ,			10_00_00_00     ,			"grey"		)
			__UpperCamelCase    :    Union[str, Any]								=      Body(0     ,			0     ,			0     ,			_lowerCAmelCase     ,			_lowerCAmelCase     ,			50_00_00_00     ,			"blue"		)
			return BodySystem([earth, moon]     ,			_lowerCAmelCase     ,			time_factor=1_00_00_00		)







def  UpperCAmelCase_   ():
			__UpperCamelCase    :    Dict								=      []
			for _ in range(10		):
						__UpperCamelCase    :    Dict								=      random.uniform(-0.5     ,			0.5		)
						__UpperCamelCase    :    str								=      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(),
					    -430000000,
					    430000000,
					    -430000000,
					    430000000,
					)
					plot("Random system with many bodies", example_a(), -1.5, 1.5, -1.5, 1.5)


import numpy as np

lowercase     :       Union[str, Any]        =       [
    ["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 		SCREAMING_SNAKE_CASE__					:







						"""simple docstring"""




						def __init__(				self    )							->	None:




									'''simple docstring'''


									__UpperCamelCase    :    Union[str, Any]								=      np.array(__UpperCamelCase    )

						def __lowerCamelCase					(				self , __UpperCamelCase    )							->	np.ndarray:




									'''simple docstring'''


									__UpperCamelCase							,				__UpperCamelCase    :    List[Any]								=      np.where(letter == self.SQUARE    )
									__UpperCamelCase    :    Tuple								=      np.concatenate([indexa + 1, indexa + 1]    )
									return indexes

						def __lowerCamelCase					(				self , __UpperCamelCase , __UpperCamelCase    )							->	str:




									'''simple docstring'''


									return self.SQUARE[indexa - 1, indexa - 1]

						def __lowerCamelCase					(				self , __UpperCamelCase    )							->	str:




									'''simple docstring'''


									__UpperCamelCase    :    int								=      message.lower()
									__UpperCamelCase    :    List[Any]								=      message.replace("j" , "i"    )

									__UpperCamelCase    :    Dict								=      ""
									for letter_index in range(len(__UpperCamelCase    )    ):
												if message[letter_index] != " ":
															__UpperCamelCase    :    int								=      self.letter_to_numbers(message[letter_index]    )
															__UpperCamelCase    :    Optional[int]								=      encoded_message + str(numbers[0]    ) + str(numbers[1]    )
												elif message[letter_index] == " ":
															__UpperCamelCase    :    List[str]								=      encoded_message + " "

									return encoded_message




						def __lowerCamelCase					(				self , __UpperCamelCase    )							->	str:




									'''simple docstring'''


									__UpperCamelCase    :    Dict								=      message.replace(" " , "  "    )
									__UpperCamelCase    :    Optional[int]								=      ""
									for numbers_index in range(int(len(__UpperCamelCase    ) / 2    )    ):
												if message[numbers_index * 2] != " ":
															__UpperCamelCase    :    int								=      message[numbers_index * 2]
															__UpperCamelCase    :    Optional[int]								=      message[numbers_index * 2 + 1]

															__UpperCamelCase    :    str								=      self.numbers_to_letter(int(__UpperCamelCase    ) , int(__UpperCamelCase    )    )
															__UpperCamelCase    :    Optional[Any]								=      decoded_message + letter
												elif message[numbers_index * 2] == " ":
															__UpperCamelCase    :    str								=      decoded_message + " "

									return decoded_message