



from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
A__     				=	20

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





class  __lowerCAmelCase			:




				def __init__(							self						,    _snake_case						,    _snake_case						,    _snake_case						,    _snake_case						,    _snake_case = 1.0						,    _snake_case = 1.0						,    _snake_case = "blue"						,    ):



									"""simple docstring"""





									_lowerCAmelCase      =   position_x
									_lowerCAmelCase      =   position_y
									_lowerCAmelCase      =   velocity_x
									_lowerCAmelCase      =   velocity_y
									_lowerCAmelCase      =   mass
									_lowerCAmelCase      =   size
									_lowerCAmelCase      =   color




				@property
				def 							snake_case       (							self		):



									"""simple docstring"""





									return self.position_x, self.position_y




				@property
				def 							snake_case       (							self		):



									"""simple docstring"""





									return self.velocity_x, self.velocity_y




				def 							snake_case       (							self						,    _snake_case						,    _snake_case						,    _snake_case		):



									"""simple docstring"""





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




				def 							snake_case       (							self						,    _snake_case		):



									"""simple docstring"""





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





class  __lowerCAmelCase			:




				def __init__(							self						,    _snake_case						,    _snake_case = 1.0						,    _snake_case = 1.0						,    _snake_case = 0.0						,    ):



									"""simple docstring"""





									_lowerCAmelCase      =   bodies
									_lowerCAmelCase      =   gravitation_constant
									_lowerCAmelCase      =   time_factor
									_lowerCAmelCase      =   softening_factor




				def __len__(							self		):



									"""simple docstring"""





									return len(self.bodies		)




				def 							snake_case       (							self						,    _snake_case		):



									"""simple docstring"""





									for bodya in self.bodies:
														_lowerCAmelCase      =   0.0
														_lowerCAmelCase      =   0.0
														for bodya in self.bodies:
																			if bodya != bodya:
																								_lowerCAmelCase      =   bodya.position_x - bodya.position_x
																								_lowerCAmelCase      =   bodya.position_y - bodya.position_y

																								# Calculation of the distance using Pythagoras's theorem
																								# Extra factor due to the softening technique
																								_lowerCAmelCase      =   (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 		_UpperCAmelCase    (				snake_case     ,						snake_case     ,						snake_case      ):





					"""simple docstring"""

					body_system.update_system(snake_case      )

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


def 		_UpperCAmelCase    (				snake_case     ,						snake_case     ,						snake_case = -1     ,						snake_case = 1     ,						snake_case = -1     ,						snake_case = 1     ,						):





					"""simple docstring"""

					_lowerCAmelCase      =   plt.figure()
					fig.canvas.set_window_title(snake_case      )
					_lowerCAmelCase      =   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
					_lowerCAmelCase      =   [
					    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(snake_case      )

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

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

					plt.show()


def 		_UpperCAmelCase    (				):





					"""simple docstring"""

					_lowerCAmelCase      =   0.9_700_436
					_lowerCAmelCase      =   -0.24_308_753
					_lowerCAmelCase      =   0.466_203_685
					_lowerCAmelCase      =   0.43_236_573

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


def 		_UpperCAmelCase    (				):





					"""simple docstring"""

					_lowerCAmelCase      =   7.34_76E22
					_lowerCAmelCase      =   5.9_72E24
					_lowerCAmelCase      =   10_22
					_lowerCAmelCase      =   3_84_39_90_00
					_lowerCAmelCase      =   6.6_74E-11

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

					_lowerCAmelCase      =   Body(-earth_moon_distance     ,						0     ,						0     ,						snake_case     ,						snake_case     ,						10_00_00_00     ,						"""grey"""      )
					_lowerCAmelCase      =   Body(0     ,						0     ,						0     ,						snake_case     ,						snake_case     ,						50_00_00_00     ,						"""blue"""      )
					return BodySystem([earth, moon]     ,						snake_case     ,						time_factor=1_00_00_00      )


def 		_UpperCAmelCase    (				):





					"""simple docstring"""

					_lowerCAmelCase      =   []
					for _ in range(10      ):
										_lowerCAmelCase      =   random.uniform(-0.5     ,						0.5      )
										_lowerCAmelCase      =   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      )     ,						snake_case     ,						snake_case     ,						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(snake_case     ,						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)








from math import exp


def 		_UpperCAmelCase    (				snake_case      ):





					"""simple docstring"""

					return 8 * x - 2 * exp(-x      )


def 		_UpperCAmelCase    (				snake_case     ,						snake_case     ,						snake_case      ):





					"""simple docstring"""

					_lowerCAmelCase      =   lower_bound
					_lowerCAmelCase      =   upper_bound
					for _ in range(0     ,						snake_case      ):
										_lowerCAmelCase     ,     _lowerCAmelCase      =   xa, xa - (f(snake_case      ) * (xa - xa)) / (f(snake_case      ) - f(snake_case      ))
					return xa


if __name__ == "__main__":
	print(f"Example: {secant_method(1, 3, 2)}")



