


'''simple docstring'''


from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
UpperCAmelCase_		:       Any							       =		20

# Time between time steps in seconds
UpperCAmelCase_		:       Union[str, Any]							       =		INTERVAL / 1000





class    lowercase__  :





							'''simple docstring'''


							def __init__(   self   , __snake_case   , __snake_case   , __snake_case   , __snake_case   , __snake_case = 1.0   , __snake_case = 1.0   , __snake_case = "blue"   , ):
											_SCREAMING_SNAKE_CASE			:	Tuple							=				position_x
											_SCREAMING_SNAKE_CASE			:	int							=				position_y
											_SCREAMING_SNAKE_CASE			:	Dict							=				velocity_x
											_SCREAMING_SNAKE_CASE			:	int							=				velocity_y
											_SCREAMING_SNAKE_CASE			:	Any							=				mass
											_SCREAMING_SNAKE_CASE			:	List[str]							=				size
											_SCREAMING_SNAKE_CASE			:	Union[str, Any]							=				color






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






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






							def UpperCAmelCase_     (   self   , __snake_case   , __snake_case   , __snake_case  ):
											self.velocity_x += force_x * delta_time
											self.velocity_y += force_y * delta_time






							def UpperCAmelCase_     (   self   , __snake_case  ):
											self.position_x += self.velocity_x * delta_time
											self.position_y += self.velocity_y * delta_time





class    lowercase__  :





							'''simple docstring'''


							def __init__(   self   , __snake_case   , __snake_case = 1.0   , __snake_case = 1.0   , __snake_case = 0.0   , ):
											_SCREAMING_SNAKE_CASE			:	Union[str, Any]							=				bodies
											_SCREAMING_SNAKE_CASE			:	int							=				gravitation_constant
											_SCREAMING_SNAKE_CASE			:	Optional[Any]							=				time_factor
											_SCREAMING_SNAKE_CASE			:	str							=				softening_factor






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






							def UpperCAmelCase_     (   self   , __snake_case  ):





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

																							# Calculation of the distance using Pythagoras's theorem
																							# Extra factor due to the softening technique
																							_SCREAMING_SNAKE_CASE			:	int							=				(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_       (  SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__					):







				"""simple docstring"""





				body_system.update_system(SCREAMING_SNAKE_CASE_					)

				# Update the positions of the patches
				for patch, body in zip(SCREAMING_SNAKE_CASE_							,				body_system.bodies					):
								_SCREAMING_SNAKE_CASE			:	Any							=				(body.position_x, body.position_y)







def       snake_case_       (  SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__ = -1							,				SCREAMING_SNAKE_CASE__ = 1							,				SCREAMING_SNAKE_CASE__ = -1							,				SCREAMING_SNAKE_CASE__ = 1							,				):







				"""simple docstring"""





				_SCREAMING_SNAKE_CASE			:	int							=				plt.figure()
				fig.canvas.set_window_title(SCREAMING_SNAKE_CASE_					)
				_SCREAMING_SNAKE_CASE			:	str							=				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
				_SCREAMING_SNAKE_CASE			:	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(SCREAMING_SNAKE_CASE_					)

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

				_SCREAMING_SNAKE_CASE			:	List[str]							=				animation.FuncAnimation(  # noqa: F841
				    SCREAMING_SNAKE_CASE_							,				SCREAMING_SNAKE_CASE_							,				interval=SCREAMING_SNAKE_CASE_							,				blit=SCREAMING_SNAKE_CASE_					)

				plt.show()







def       snake_case_       (  ):







				"""simple docstring"""





				_SCREAMING_SNAKE_CASE			:	Any							=				0.9_7_0_0_4_3_6
				_SCREAMING_SNAKE_CASE			:	List[str]							=				-0.2_4_3_0_8_7_5_3
				_SCREAMING_SNAKE_CASE			:	List[Any]							=				0.4_6_6_2_0_3_6_8_5
				_SCREAMING_SNAKE_CASE			:	Tuple							=				0.4_3_2_3_6_5_7_3

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







def       snake_case_       (  ):







				"""simple docstring"""





				_SCREAMING_SNAKE_CASE			:	List[Any]							=				7.34_76E22
				_SCREAMING_SNAKE_CASE			:	Dict							=				5.9_72E24
				_SCREAMING_SNAKE_CASE			:	Any							=				1022
				_SCREAMING_SNAKE_CASE			:	int							=				3_8439_9000
				_SCREAMING_SNAKE_CASE			:	str							=				6.6_74E-11

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

				_SCREAMING_SNAKE_CASE			:	int							=				Body(-earth_moon_distance							,				0							,				0							,				SCREAMING_SNAKE_CASE_							,				SCREAMING_SNAKE_CASE_							,				1000_0000							,				"""grey"""					)
				_SCREAMING_SNAKE_CASE			:	Any							=				Body(0							,				0							,				0							,				SCREAMING_SNAKE_CASE_							,				SCREAMING_SNAKE_CASE_							,				5000_0000							,				"""blue"""					)
				return BodySystem([earth, moon]							,				SCREAMING_SNAKE_CASE_							,				time_factor=100_0000					)







def       snake_case_       (  ):







				"""simple docstring"""





				_SCREAMING_SNAKE_CASE			:	Optional[int]							=				[]
				for _ in range(10					):
								_SCREAMING_SNAKE_CASE			:	Optional[int]							=				random.uniform(-0.5							,				0.5					)
								_SCREAMING_SNAKE_CASE			:	Optional[int]							=				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					)							,				SCREAMING_SNAKE_CASE_							,				SCREAMING_SNAKE_CASE_							,				size=0.0_5							,				)					)
								bodies.append(
								    Body(
								        random.uniform(-0.5							,				0.5					)							,				random.uniform(-0.5							,				0.5					)							,				-velocity_x							,				-velocity_y							,				size=0.0_5							,				)					)
				return BodySystem(SCREAMING_SNAKE_CASE_							,				0.0_1							,				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_3000_0000,
					    4_3000_0000,
					    -4_3000_0000,
					    4_3000_0000,
					)
					plot('Random system with many bodies', example_a(), -1.5, 1.5, -1.5, 1.5)








'''simple docstring'''


import sys







def       snake_case_       (  SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__					):







				"""simple docstring"""





				if not numbers:
								return 0
				if target == 0 and target in numbers:
								return 0
				if not isinstance(SCREAMING_SNAKE_CASE__							,				(list, tuple)					) or not all(
				    isinstance(SCREAMING_SNAKE_CASE__							,				SCREAMING_SNAKE_CASE__					) for number in numbers					):
								raise ValueError("""numbers must be an iterable of integers"""					)

				_SCREAMING_SNAKE_CASE			:	str							=				0
				_SCREAMING_SNAKE_CASE			:	int							=				sys.maxsize

				while right < len(SCREAMING_SNAKE_CASE__					):
								curr_sum += numbers[right]
								while curr_sum >= target and left <= right:
												_SCREAMING_SNAKE_CASE			:	Any							=				min(SCREAMING_SNAKE_CASE__							,				right - left + 1					)
												curr_sum -= numbers[left]
												left += 1
								right += 1

				return 0 if min_len == sys.maxsize else min_len




