


"""simple docstring"""






from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
SCREAMING_SNAKE_CASE__				:  Any						= 20

# Time between time steps in seconds
SCREAMING_SNAKE_CASE__				:  int						= INTERVAL / 1000


class 	A_       :





  """simple docstring"""

  def __init__(						self  ,					__UpperCAmelCase  ,					__UpperCAmelCase  ,					__UpperCAmelCase  ,					__UpperCAmelCase  ,					__UpperCAmelCase = 1.0  ,					__UpperCAmelCase = 1.0  ,					__UpperCAmelCase = "blue"  ,					)				->					Optional[Any]:
     a   :	Tuple						=    position_x
     a   :	Any						=    position_y
     a   :	List[str]						=    velocity_x
     a   :	Optional[Any]						=    velocity_y
     a   :	Any						=    mass
     a   :	Tuple						=    size
     a   :	Tuple						=    color
  @property
  def 							lowercase_     (						self						)				->					int:
     return self.position_x, self.position_y
  @property
  def 							lowercase_     (						self						)				->					List[str]:
     return self.velocity_x, self.velocity_y
  def 							lowercase_     (						self  ,					__UpperCAmelCase  ,					__UpperCAmelCase  ,					__UpperCAmelCase						)				->					Any:
     self.velocity_x += force_x * delta_time
     self.velocity_y += force_y * delta_time






  def 							lowercase_     (						self  ,					__UpperCAmelCase						)				->					List[Any]:
     self.position_x += self.velocity_x * delta_time
     self.position_y += self.velocity_y * delta_time



class 	A_       :





  """simple docstring"""

  def __init__(						self  ,					__UpperCAmelCase  ,					__UpperCAmelCase = 1.0  ,					__UpperCAmelCase = 1.0  ,					__UpperCAmelCase = 0.0  ,					)				->					int:
     a   :	Tuple						=    bodies
     a   :	List[Any]						=    gravitation_constant
     a   :	List[str]						=    time_factor
     a   :	Union[str, Any]						=    softening_factor
  def __len__(						self						)				->					Optional[int]:
     return len(self.bodies						)






  def 							lowercase_     (						self  ,					__UpperCAmelCase						)				->					Dict:





     for bodya in self.bodies:
        a   :	int						=    0.0
        a   :	Optional[int]						=    0.0
        for bodya in self.bodies:
           if bodya != bodya:
              a   :	Optional[Any]						=    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   :	str						=    (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        A_					(					UpperCAmelCase__       ,   UpperCAmelCase__       ,   UpperCAmelCase__				)     ->     None:
   body_system.update_system(a_				)

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




def        A_					(					UpperCAmelCase__       ,   UpperCAmelCase__       ,   UpperCAmelCase__ = -1       ,   UpperCAmelCase__ = 1       ,   UpperCAmelCase__ = -1       ,   UpperCAmelCase__ = 1       ,   )     ->     None:
   a   :	Union[str, Any]						=    plt.figure()
   fig.canvas.set_window_title(a_				)
   a   :	Union[str, 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
   a   :	Union[str, 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(a_				)

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

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

   plt.show()




def        A_					(					)     ->     BodySystem:
   a   :	Any						=    0.9_7_0_0_4_3_6
   a   :	Dict						=    -0.2_4_3_0_8_7_5_3
   a   :	Tuple						=    0.4_6_6_2_0_3_6_8_5
   a   :	List[str]						=    0.4_3_2_3_6_5_7_3

   a   :	List[str]						=    [
       Body(a_       ,   a_       ,   a_       ,   a_       ,   size=0.2       ,   color='red'				),
       Body(-position_x       ,   -position_y       ,   a_       ,   a_       ,   size=0.2       ,   color='green'				),
       Body(0       ,   0       ,   -2 * velocity_x       ,   -2 * velocity_y       ,   size=0.2       ,   color='blue'				),
   ]
   return BodySystem(a_       ,   time_factor=3				)




def        A_					(					)     ->     BodySystem:
   a   :	Optional[Any]						=    7.34_76e22
   a   :	str						=    5.9_72e24
   a   :	str						=    1022
   a   :	str						=    3_8439_9000
   a   :	Optional[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
   a   :	Optional[int]						=    earth_mass * velocity_dif / (earth_mass + moon_mass)
   a   :	Any						=    moon_velocity - velocity_dif

   a   :	Optional[int]						=    Body(-earth_moon_distance       ,   0       ,   0       ,   a_       ,   a_       ,   1000_0000       ,   'grey'				)
   a   :	Optional[int]						=    Body(0       ,   0       ,   0       ,   a_       ,   a_       ,   5000_0000       ,   'blue'				)
   return BodySystem([earth, moon]       ,   a_       ,   time_factor=100_0000				)




def        A_					(					)     ->     BodySystem:
   a   :	Tuple						=    []
   for _ in range(10				):
      a   :	Any						=    random.uniform(-0.5       ,   0.5				)
      a   :	List[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				)       ,   a_       ,   a_       ,   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(a_       ,   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)







from __future__ import annotations

import math
import random

def    UpperCamelCase_ (	a_  ,				a_  ,				a_		)     ->list[int]:
		A				=d * e - 1
		A				=0
		A				=0
		while p == 0:
				A				=random.randint(2  ,				n - 1		)
				A				=k
				while True:
						if t % 2 == 0:
								A				=t // 2
								A				=(g**t) % n
								A				=math.gcd(x - 1  ,				a_		)
								if x > 1 and y > 1:
										A				=y
										A				=n // y
										break  # find the correct factors
						else:
								break  # t is not divisible by 2, break and choose another g
		return sorted([p, q]		)


if __name__ == "__main__":
	import doctest

	doctest.testmod()




