





from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
UpperCamelCase__          =20

# Time between time steps in seconds
UpperCamelCase__          =INTERVAL / 1000





class 	lowerCAmelCase__:






       '''simple docstring'''


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




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




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




       def 			UpperCamelCase_	(					self			,  __lowerCamelCase			,  __lowerCamelCase			,  __lowerCamelCase							)	->   None:
            self.velocity_x += force_x * delta_time
            self.velocity_y += force_y * delta_time







       def 			UpperCamelCase_	(					self			,  __lowerCamelCase							)	->   None:
            self.position_x += self.velocity_x * delta_time
            self.position_y += self.velocity_y * delta_time





class 	lowerCAmelCase__:






       '''simple docstring'''


       def __init__(					self			,  __lowerCamelCase			,  __lowerCamelCase = 1.0			,  __lowerCamelCase = 1.0			,  __lowerCamelCase = 0.0			,  )	->   None:
            _SCREAMING_SNAKE_CASE				:		List[str]  			=							bodies
            _SCREAMING_SNAKE_CASE				:		Tuple  			=							gravitation_constant
            _SCREAMING_SNAKE_CASE				:		Dict  			=							time_factor
            _SCREAMING_SNAKE_CASE				:		Dict  			=							softening_factor




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







       def 			UpperCamelCase_	(					self			,  __lowerCamelCase							)	->   None:
            for bodya in self.bodies:
                 _SCREAMING_SNAKE_CASE				:		Tuple  			=							0.0
                 _SCREAMING_SNAKE_CASE				:		Any  			=							0.0
                 for bodya in self.bodies:
                      if bodya != bodya:
                           _SCREAMING_SNAKE_CASE				:		Optional[Any]  			=							bodya.position_x - bodya.position_x
                           _SCREAMING_SNAKE_CASE				:		Union[str, 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				:		List[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(__lowerCamelCase			,  __lowerCamelCase			,  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 			lowerCamelCase__	(__lowerCamelCase,				__lowerCamelCase,				__lowerCamelCase		):
     body_system.update_system(__lowerCamelCase		)

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






def 			lowerCamelCase__	(__lowerCamelCase,				__lowerCamelCase,				__lowerCamelCase = -1,				__lowerCamelCase = 1,				__lowerCamelCase = -1,				__lowerCamelCase = 1,				):
     _SCREAMING_SNAKE_CASE				:		Any  			=							plt.figure()
     fig.canvas.set_window_title(__lowerCamelCase		)
     _SCREAMING_SNAKE_CASE				:		Optional[int]  			=							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				:		Optional[Any]  			=							[
         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		) -> list[plt.Circle]:
          update_step(__lowerCamelCase,				__lowerCamelCase,				__lowerCamelCase		)
          return patches

     _SCREAMING_SNAKE_CASE				:		Optional[int]  			=							animation.FuncAnimation(  # noqa: F841
         __lowerCamelCase,				__lowerCamelCase,				interval=__lowerCamelCase,				blit=__lowerCamelCase		)

     plt.show()






def 			lowerCamelCase__	():
     _SCREAMING_SNAKE_CASE				:		int  			=							0.970_0436
     _SCREAMING_SNAKE_CASE				:		Dict  			=							-0.2430_8753
     _SCREAMING_SNAKE_CASE				:		List[str]  			=							0.4_6620_3685
     _SCREAMING_SNAKE_CASE				:		List[Any]  			=							0.4323_6573

     _SCREAMING_SNAKE_CASE				:		int  			=							[
         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 			lowerCamelCase__	():
     _SCREAMING_SNAKE_CASE				:		List[str]  			=							7.3476e22
     _SCREAMING_SNAKE_CASE				:		str  			=							5.972e24
     _SCREAMING_SNAKE_CASE				:		int  			=							1022
     _SCREAMING_SNAKE_CASE				:		Union[str, Any]  			=							384399000
     _SCREAMING_SNAKE_CASE				:		Union[str, Any]  			=							6.674e-11

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

     _SCREAMING_SNAKE_CASE				:		Optional[int]  			=							Body(-earth_moon_distance,				0,				0,				__lowerCamelCase,				__lowerCamelCase,				10000000,				"grey"		)
     _SCREAMING_SNAKE_CASE				:		Dict  			=							Body(0,				0,				0,				__lowerCamelCase,				__lowerCamelCase,				50000000,				"blue"		)
     return BodySystem([earth, moon],				__lowerCamelCase,				time_factor=1000000		)






def 			lowerCamelCase__	():
     _SCREAMING_SNAKE_CASE				:		Optional[int]  			=							[]
     for _ in range(10		):
          _SCREAMING_SNAKE_CASE				:		Optional[Any]  			=							random.uniform(-0.5,				0.5		)
          _SCREAMING_SNAKE_CASE				:		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		),				__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(),
           -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'''







from __future__ import annotations






def       lowerCAmelCase__       (				lowerCamelCase     :			list   ):
       _A     :	Union[str, Any]      =					sorted(lowerCamelCase   )
       _A     :	Tuple      =					len(lowerCamelCase   )
       _A     :	int      =					length >> 1
       return (
           (sorted_list[mid_index] + sorted_list[mid_index - 1]) / 2
           if length % 2 == 0
           else sorted_list[mid_index]
       )






def       lowerCAmelCase__       (				):
       import doctest

       doctest.testmod()


if __name__ == "__main__":
   main()






