




from __future__ import annotations

import random

from matplotlib import animation
from matplotlib import pyplot as plt

# Frame rate of the animation
SCREAMING_SNAKE_CASE__	:				List[str]  =       20

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




class 		UpperCamelCase__     :

    '''simple docstring'''



    def __init__(   self					,  UpperCamelCase__					,  UpperCamelCase__					,  UpperCamelCase__					,  UpperCamelCase__					,  UpperCamelCase__ = 1.0					,  UpperCamelCase__ = 1.0					,  UpperCamelCase__ = "blue"					,  )					->				List[Any]:
           lowerCamelCase       :				Dict      			= position_x
           lowerCamelCase       :				Union[str, Any]      			= position_y
           lowerCamelCase       :				Union[str, Any]      			= velocity_x
           lowerCamelCase       :				List[str]      			= velocity_y
           lowerCamelCase       :				List[Any]      			= mass
           lowerCamelCase       :				Union[str, Any]      			= size
           lowerCamelCase       :				Dict      			= color







    @property
    def 	_lowercase					(   self					)					->				Any:
           return self.position_x, self.position_y







    @property
    def 	_lowercase					(   self					)					->				int:
           return self.velocity_x, self.velocity_y







    def 	_lowercase					(   self					,  UpperCamelCase__					,  UpperCamelCase__					,  UpperCamelCase__					)					->				Tuple:
           self.velocity_x += force_x * delta_time
           self.velocity_y += force_y * delta_time







    def 	_lowercase					(   self					,  UpperCamelCase__					)					->				Any:
           self.position_x += self.velocity_x * delta_time
           self.position_y += self.velocity_y * delta_time




class 		UpperCamelCase__     :

    '''simple docstring'''



    def __init__(   self					,  UpperCamelCase__					,  UpperCamelCase__ = 1.0					,  UpperCamelCase__ = 1.0					,  UpperCamelCase__ = 0.0					,  )					->				Union[str, Any]:
           lowerCamelCase       :				int      			= bodies
           lowerCamelCase       :				Optional[Any]      			= gravitation_constant
           lowerCamelCase       :				Optional[int]      			= time_factor
           lowerCamelCase       :				Any      			= softening_factor







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







    def 	_lowercase					(   self					,  UpperCamelCase__					)					->				List[str]:







           for bodya in self.bodies:
                  lowerCamelCase       :				Any      			= 0.0
                  lowerCamelCase       :				int      			= 0.0
                  for bodya in self.bodies:
                         if bodya != bodya:
                                lowerCamelCase       :				Optional[int]      			= bodya.position_x - bodya.position_x
                                lowerCamelCase       :				str      			= bodya.position_y - bodya.position_y

                                # Calculation of the distance using Pythagoras's theorem
                                # Extra factor due to the softening technique
                                lowerCamelCase       :				Union[str, 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(UpperCamelCase__					,  UpperCamelCase__					,  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				(				_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE   )     ->			None:
       body_system.update_system(_SCREAMING_SNAKE_CASE   )

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

def 					A				(				_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE = -1			,_SCREAMING_SNAKE_CASE = 1			,_SCREAMING_SNAKE_CASE = -1			,_SCREAMING_SNAKE_CASE = 1			,)     ->			None:
       lowerCamelCase       :				Any      			= plt.figure()
       fig.canvas.set_window_title(_SCREAMING_SNAKE_CASE   )
       lowerCamelCase       :				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
       lowerCamelCase       :				List[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

       lowerCamelCase       :				int      			= animation.FuncAnimation(  # noqa: F841
           _SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE			,interval=_SCREAMING_SNAKE_CASE			,blit=_SCREAMING_SNAKE_CASE   )

       plt.show()

def 					A				(				)     ->			BodySystem:
       lowerCamelCase       :				Tuple      			= 0.9700436
       lowerCamelCase       :				Tuple      			= -0.24308753
       lowerCamelCase       :				List[Any]      			= 0.466203685
       lowerCamelCase       :				Union[str, Any]      			= 0.43236573

       lowerCamelCase       :				Any      			= [
           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 					A				(				)     ->			BodySystem:
       lowerCamelCase       :				Dict      			= 7.34_76e22
       lowerCamelCase       :				Optional[Any]      			= 5.9_72e24
       lowerCamelCase       :				Optional[Any]      			= 1022
       lowerCamelCase       :				Tuple      			= 3_8439_9000
       lowerCamelCase       :				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
       lowerCamelCase       :				Any      			= earth_mass * velocity_dif / (earth_mass + moon_mass)
       lowerCamelCase       :				List[Any]      			= moon_velocity - velocity_dif

       lowerCamelCase       :				Tuple      			= Body(-earth_moon_distance			,0			,0			,_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE			,1000_0000			,"grey"   )
       lowerCamelCase       :				Optional[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 					A				(				)     ->			BodySystem:
       lowerCamelCase       :				Dict      			= []
       for _ in range(10   ):
              lowerCamelCase       :				Dict      			= random.uniform(-0.5			,0.5   )
              lowerCamelCase       :				Optional[Any]      			= 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=0.05			,)   )
              bodies.append(
                  Body(
                      random.uniform(-0.5			,0.5   )			,random.uniform(-0.5			,0.5   )			,-velocity_x			,-velocity_y			,size=0.05			,)   )
       return BodySystem(_SCREAMING_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(),
           -430000000,
           430000000,
           -430000000,
           430000000,
       )
       plot('Random system with many bodies', example_a(), -1.5, 1.5, -1.5, 1.5)












import numpy as np

def 					A				(				_SCREAMING_SNAKE_CASE   )     ->			str:
       lowerCamelCase			,				lowerCamelCase       :				Dict      			= a.shape
       lowerCamelCase       :				Union[str, Any]      			= min(_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE   )
       lowerCamelCase       :				Union[str, Any]      			= np.eye(_SCREAMING_SNAKE_CASE   )
       lowerCamelCase       :				List[Any]      			= a.copy()

       for k in range(t - 1   ):
              # select a column of modified matrix A':
              lowerCamelCase       :				Optional[Any]      			= r[k:, [k]]
              # construct first basis vector
              lowerCamelCase       :				Union[str, Any]      			= np.zeros_like(_SCREAMING_SNAKE_CASE   )
              lowerCamelCase       :				int      			= 1.0
              # determine scaling factor
              lowerCamelCase       :				Any      			= np.linalg.norm(_SCREAMING_SNAKE_CASE   )
              # construct vector v for Householder reflection
              lowerCamelCase       :				List[str]      			= x + np.sign(x[0]   ) * alpha * ea
              v /= np.linalg.norm(_SCREAMING_SNAKE_CASE   )

              # construct the Householder matrix
              lowerCamelCase       :				List[Any]      			= np.eye(m - k   ) - 2.0 * v @ v.T
              # pad with ones and zeros as necessary
              lowerCamelCase       :				Union[str, Any]      			= np.block([[np.eye(_SCREAMING_SNAKE_CASE   ), np.zeros((k, m - k)   )], [np.zeros((m - k, k)   ), q_k]]   )

              lowerCamelCase       :				Any      			= q @ q_k.T
              lowerCamelCase       :				List[Any]      			= q_k @ r

       return q, r


if __name__ == "__main__":
       import doctest

       doctest.testmod()






