






"""simple docstring"""



from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray



def 						lowercase    (				__snake_case				: str		,					__snake_case				: List[Any]		,					__snake_case				: str = False	):


		if radian_mode:
				return [magnitude * cos(lowerCAmelCase_	), magnitude * sin(lowerCAmelCase_	)]
		return [magnitude * cos(radians(lowerCAmelCase_	)	), magnitude * sin(radians(lowerCAmelCase_	)	)]



def 						lowercase    (				__snake_case				: int		,					__snake_case				: Union[str, Any]		,					__snake_case				: Optional[Any] = 1_0**-1	):
		lowercase_ : NDArray[floataa]							       =   cross(lowerCAmelCase_		,					lowerCAmelCase_	)
		lowercase_ : float							       =   sum(lowerCAmelCase_	)
		return abs(lowerCAmelCase_	) < eps


if __name__ == "__main__":
						# Test to check if it works
						__A		:       str  				=		array(
						    [
						        polar_force(718.4, 180 - 30),
						        polar_force(879.54, 45),
						        polar_force(100, -90),
						    ]
						)

						__A		:       NDArray[floataa]  				=		array([[0, 0], [0, 0], [0, 0]])

						assert in_static_equilibrium(forces, location)

						# Problem 1 in image_data/2D_problems.jpg
						__A		:       List[str]  				=		array(
						    [
						        polar_force(30 * 9.81, 15),
						        polar_force(215, 180 - 45),
						        polar_force(264, 90 - 30),
						    ]
						)

						__A		:       Tuple  				=		array([[0, 0], [0, 0], [0, 0]])

						assert in_static_equilibrium(forces, location)

						# Problem in image_data/2D_problems_1.jpg
						__A		:       Dict  				=		array([[0, -2_000], [0, -1_200], [0, 15_600], [0, -12_400]])

						__A		:       Union[str, Any]  				=		array([[0, 0], [6, 0], [10, 0], [12, 0]])

						assert in_static_equilibrium(forces, location)

						import doctest

						doctest.testmod()








'''simple docstring'''






class 							lowercase       :


 """simple docstring"""


 def __init__(							self							)		->   List[str]:
        _UpperCAmelCase  :						int			     = 0
        _UpperCAmelCase  :						Union[str, Any]			     = 0
        _UpperCAmelCase  :						Optional[int]			     = {}







 def _snake_case (							self  ,a_							)		->   Optional[Any]:
        if vertex not in self.adjacency:
               _UpperCAmelCase  :						int			     = {}
               self.num_vertices += 1







 def _snake_case (							self  ,a_  ,a_  ,a_							)		->   int:
        self.add_vertex(a_							)
        self.add_vertex(a_							)

        if head == tail:
               return

        _UpperCAmelCase  :						List[Any]			     = weight
        _UpperCAmelCase  :						Dict			     = weight







 def _snake_case (							self							)		->   Dict:
        _UpperCAmelCase  :						Optional[int]			     = self.get_edges()
        for edge in edges:
               _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						Dict			     = edge
               edges.remove((tail, head, weight)							)
        for i in range(len(a_							)							):
               _UpperCAmelCase  :						str			     = list(edges[i]							)

        edges.sort(key=lambda a_							: e[2]							)
        for i in range(len(a_							) - 1							):
               if edges[i][2] >= edges[i + 1][2]:
                      _UpperCAmelCase  :						Optional[Any]			     = edges[i][2] + 1
        for edge in edges:
               _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						Union[str, Any]			     = edge
               _UpperCAmelCase  :						str			     = weight
               _UpperCAmelCase  :						List[str]			     = weight







 def __str__(							self							)		->   Any:
        _UpperCAmelCase  :						List[Any]			     = """"""
        for tail in self.adjacency:
               for head in self.adjacency[tail]:
                      _UpperCAmelCase  :						List[str]			     = self.adjacency[head][tail]
                      string += f'''{head} -> {tail} == {weight}\n'''
        return string.rstrip("""\n"""							)







 def _snake_case (							self							)		->   str:
        _UpperCAmelCase  :						int			     = []
        for tail in self.adjacency:
               for head in self.adjacency[tail]:
                      output.append((tail, head, self.adjacency[head][tail])							)
        return output







 def _snake_case (							self							)		->   Optional[int]:
        return self.adjacency.keys()







 @staticmethod
 def _snake_case (							a_=None  ,a_=None							)		->   Tuple:
        _UpperCAmelCase  :						List[Any]			     = Graph()
        if vertices is None:
               _UpperCAmelCase  :						List[str]			     = []
        if edges is None:
               _UpperCAmelCase  :						Optional[Any]			     = []
        for vertex in vertices:
               g.add_vertex(a_							)
        for edge in edges:
               g.add_edge(*a_							)
        return g



 class 							lowercase       :


  """simple docstring"""


  def __init__(							self							)		->   int:
         _UpperCAmelCase  :						List[str]			     = {}
         _UpperCAmelCase  :						int			     = {}







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







  def _snake_case (							self  ,a_							)		->   str:
         if item in self.parent:
                return self.find(a_							)

         _UpperCAmelCase  :						Optional[Any]			     = item
         _UpperCAmelCase  :						List[Any]			     = 0
         return item







  def _snake_case (							self  ,a_							)		->   List[str]:
         if item not in self.parent:
                return self.make_set(a_							)
         if item != self.parent[item]:
                _UpperCAmelCase  :						List[Any]			     = self.find(self.parent[item]							)
         return self.parent[item]







  def _snake_case (							self  ,a_  ,a_							)		->   Union[str, Any]:
         _UpperCAmelCase  :						Any			     = self.find(a_							)
         _UpperCAmelCase  :						List[str]			     = self.find(a_							)

         if roota == roota:
                return roota

         if self.rank[roota] > self.rank[roota]:
                _UpperCAmelCase  :						Any			     = roota
                return roota

         if self.rank[roota] < self.rank[roota]:
                _UpperCAmelCase  :						Any			     = roota
                return roota

         if self.rank[roota] == self.rank[roota]:
                self.rank[roota] += 1
                _UpperCAmelCase  :						List[str]			     = roota
                return roota
         return None







 @staticmethod
 def _snake_case (							a_							)		->   List[Any]:
        _UpperCAmelCase  :						int			     = graph.num_vertices

        _UpperCAmelCase  :						int			     = Graph.UnionFind()
        _UpperCAmelCase  :						Optional[int]			     = []
        while num_components > 1:
               _UpperCAmelCase  :						int			     = {}
               for vertex in graph.get_vertices():
                      _UpperCAmelCase  :						Union[str, Any]			     = -1

               _UpperCAmelCase  :						Tuple			     = graph.get_edges()
               for edge in edges:
                      _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						str			     = edge
                      edges.remove((tail, head, weight)							)
               for edge in edges:
                      _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						Optional[Any]			     = edge
                      _UpperCAmelCase  :						Any			     = union_find.find(a_							)
                      _UpperCAmelCase  :						Any			     = union_find.find(a_							)
                      if seta != seta:
                             if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
                                    _UpperCAmelCase  :						Tuple			     = [head, tail, weight]

                             if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
                                    _UpperCAmelCase  :						List[str]			     = [head, tail, weight]
               for vertex in cheap_edge:
                      if cheap_edge[vertex] != -1:
                             _UpperCAmelCase     ,_UpperCAmelCase     ,_UpperCAmelCase  :						str			     = cheap_edge[vertex]
                             if union_find.find(a_							) != union_find.find(a_							):
                                    union_find.union(a_  ,a_							)
                                    mst_edges.append(cheap_edge[vertex]							)
                                    _UpperCAmelCase  :						Tuple			     = num_components - 1
        _UpperCAmelCase  :						Optional[int]			     = Graph.build(edges=a_							)
        return mst



