



from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
snake_case     :    Tuple						=    2_0

# Time between time steps in seconds
snake_case     :    Union[str, Any]						=    INTERVAL / 1_0_0_0




class 				lowerCAmelCase__	:







					def __init__(							self   :  Tuple      ,		_A   :  float      ,		_A   :  float      ,		_A   :  float      ,		_A   :  float      ,		_A   :  float = 1.0      ,		_A   :  float = 1.0      ,		_A   :  str = "blue"      ,		):
											A__						:				Any         =			position_x
											A__						:				List[str]         =			position_y
											A__						:				Optional[int]         =			velocity_x
											A__						:				List[Any]         =			velocity_y
											A__						:				List[str]         =			mass
											A__						:				Optional[Any]         =			size
											A__						:				Optional[Any]         =			color







					@property
					def 			_lowercase      (							self   :  str):
											return self.position_x, self.position_y







					@property
					def 			_lowercase      (							self   :  Dict):
											return self.velocity_x, self.velocity_y







					def 			_lowercase      (							self   :  str      ,		_A   :  float      ,		_A   :  float      ,		_A   :  float):
											self.velocity_x += force_x * delta_time
											self.velocity_y += force_y * delta_time







					def 			_lowercase      (							self   :  Tuple      ,		_A   :  float):
											self.position_x += self.velocity_x * delta_time
											self.position_y += self.velocity_y * delta_time




class 				lowerCAmelCase__	:







					def __init__(							self   :  List[Any]      ,		_A   :  list[Body]      ,		_A   :  float = 1.0      ,		_A   :  float = 1.0      ,		_A   :  float = 0.0      ,		):
											A__						:				List[str]         =			bodies
											A__						:				Optional[Any]         =			gravitation_constant
											A__						:				Union[str, Any]         =			time_factor
											A__						:				Any         =			softening_factor







					def __len__(							self   :  Any):
											return len(self.bodies)







					def 			_lowercase      (							self   :  List[str]      ,		_A   :  float):
											for bodya in self.bodies:
																	A__						:				Optional[int]         =			0.0
																	A__						:				List[Any]         =			0.0
																	for bodya in self.bodies:
																							if bodya != bodya:
																													A__						:				Tuple         =			bodya.position_x - bodya.position_x
																													A__						:				Tuple         =			bodya.position_y - bodya.position_y

																													# Calculation of the distance using Pythagoras's theorem
																													# Extra factor due to the softening technique
																													A__						:				Tuple         =			(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(_A      ,		_A      ,		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 						snake_case__							(       __lowercase				,  __lowercase				,  __lowercase							)		->		None:







						"""simple docstring"""

						body_system.update_system(__lowercase							)

						# Update the positions of the patches
						for patch, body in zip(__lowercase				,  body_system.bodies							):
												A__						:				Any         =			(body.position_x, body.position_y)
def 						snake_case__							(       __lowercase				,  __lowercase				,  __lowercase = -1				,  __lowercase = 1				,  __lowercase = -1				,  __lowercase = 1				,  )		->		None:







						"""simple docstring"""

						A__						:				List[Any]         =			plt.figure()
						fig.canvas.set_window_title(__lowercase							)
						A__						:				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
						A__						:				Dict         =			[
						    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(__lowercase							)

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

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

						plt.show()
def 						snake_case__							(       )		->		BodySystem:







						"""simple docstring"""

						A__						:				Optional[Any]         =			0.9700436
						A__						:				Any         =			-0.24308753
						A__						:				List[str]         =			0.466203685
						A__						:				Dict         =			0.43236573

						A__						:				str         =			[
						    Body(__lowercase				,  __lowercase				,  __lowercase				,  __lowercase				,  size=0.2				,  color="red"							),
						    Body(-position_x				,  -position_y				,  __lowercase				,  __lowercase				,  size=0.2				,  color="green"							),
						    Body(0				,  0				,  -2 * velocity_x				,  -2 * velocity_y				,  size=0.2				,  color="blue"							),
						]
						return BodySystem(__lowercase				,  time_factor=3							)
def 						snake_case__							(       )		->		BodySystem:







						"""simple docstring"""

						A__						:				str         =			7.34_76E22
						A__						:				Tuple         =			5.9_72E24
						A__						:				int         =			1_0_2_2
						A__						:				Optional[Any]         =			3_8_4_3_9_9_0_0_0
						A__						:				Union[str, Any]         =			6.6_74E-11

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

						A__						:				int         =			Body(-earth_moon_distance				,  0				,  0				,  __lowercase				,  __lowercase				,  1_0_0_0_0_0_0_0				,  "grey"							)
						A__						:				Optional[Any]         =			Body(0				,  0				,  0				,  __lowercase				,  __lowercase				,  5_0_0_0_0_0_0_0				,  "blue"							)
						return BodySystem([earth, moon]				,  __lowercase				,  time_factor=1_0_0_0_0_0_0							)
def 						snake_case__							(       )		->		BodySystem:







						"""simple docstring"""

						A__						:				Tuple         =			[]
						for _ in range(1_0							):
												A__						:				str         =			random.uniform(-0.5				,  0.5							)
												A__						:				Dict         =			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							)				,  __lowercase				,  __lowercase				,  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(__lowercase				,  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_3_0_0_0_0_0_0_0,
		    4_3_0_0_0_0_0_0_0,
		    -4_3_0_0_0_0_0_0_0,
		    4_3_0_0_0_0_0_0_0,
		)
		plot('Random system with many bodies', example_a(), -1.5, 1.5, -1.5, 1.5)




from __future__ import annotations
def 						snake_case__							(       __lowercase				,  __lowercase				,  __lowercase				,  )		->		list[float | int]:







						"""simple docstring"""

						if not all((nth_term, start_term_a, common_ratio_r)							):
												return []
						A__						:				list[float | int]         =			[]
						A__						:				Optional[Any]         =			1
						A__						:				List[Any]         =			common_ratio_r
						for _ in range(int(__lowercase							)							):
												if not series:
																		series.append(__lowercase							)
												else:
																		power += 1
																		series.append(float(start_term_a * multiple							)							)
																		A__						:				Optional[Any]         =			pow(float(__lowercase							)				,  __lowercase							)
						return series


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		snake_case     :    Optional[int]						=    float(input('Enter the last number (n term) of the Geometric Series'))
		snake_case     :    str						=    float(input('Enter the starting term (a) of the Geometric Series'))
		snake_case     :    int						=    float(
		    input('Enter the common ratio between two terms (r) of the Geometric Series')
		)
		print('Formula of Geometric Series => a + ar + ar^2 ... +ar^n')
		print(geometric_series(nth_term, start_term_a, common_ratio_r))