
'''simple docstring'''



from __future__ import annotations

from math import pi, pow
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_  )   ->   Optional[int]:




   """simple docstring"""





   if side_length < 0:
      raise ValueError("""vol_cube() only accepts non-negative values"""  )
   return pow(lowerCamelCase_			,					3  )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Optional[int]:




   """simple docstring"""





   if height < 0 or radius < 0:
      raise ValueError("""vol_spherical_cap() only accepts non-negative values"""  )
   # Volume is 1/3 pi * height squared * (3 * radius - height)
   return 1 / 3 * pi * pow(lowerCamelCase_			,					2  ) * (3 * radius - height)
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   str:




   """simple docstring"""





   if radius_a < 0 or radius_a < 0 or centers_distance < 0:
      raise ValueError("""vol_spheres_intersect() only accepts non-negative values"""  )
   if centers_distance == 0:
      return vol_sphere(min(lowerCamelCase_			,					lowerCamelCase_  )  )

   _SCREAMING_SNAKE_CASE          = (
       (radius_a - radius_a + centers_distance)
       * (radius_a + radius_a - centers_distance)
       / (2 * centers_distance)
   )
   _SCREAMING_SNAKE_CASE          = (
       (radius_a - radius_a + centers_distance)
       * (radius_a + radius_a - centers_distance)
       / (2 * centers_distance)
   )

   return vol_spherical_cap(lowerCamelCase_			,					lowerCamelCase_  ) + vol_spherical_cap(lowerCamelCase_			,					lowerCamelCase_  )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   str:




   """simple docstring"""





   if radius_a <= 0 or radius_a <= 0 or centers_distance < 0:
      raise ValueError(
          """vol_spheres_union() only accepts non-negative values, non-zero radius"""  )

   if centers_distance == 0:
      return vol_sphere(max(lowerCamelCase_			,					lowerCamelCase_  )  )

   return (
       vol_sphere(lowerCamelCase_  )
       + vol_sphere(lowerCamelCase_  )
       - vol_spheres_intersect(lowerCamelCase_			,					lowerCamelCase_			,					lowerCamelCase_  )
   )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Tuple:




   """simple docstring"""





   if width < 0 or height < 0 or length < 0:
      raise ValueError("""vol_cuboid() only accepts non-negative values"""  )
   return float(width * height * length  )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Optional[Any]:




   """simple docstring"""





   if height < 0 or area_of_base < 0:
      raise ValueError("""vol_cone() only accepts non-negative values"""  )
   return area_of_base * height / 3.0
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Optional[int]:




   """simple docstring"""





   if height < 0 or radius < 0:
      raise ValueError("""vol_right_circ_cone() only accepts non-negative values"""  )
   return pi * pow(lowerCamelCase_			,					2  ) * height / 3.0
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   str:




   """simple docstring"""





   if height < 0 or area_of_base < 0:
      raise ValueError("""vol_prism() only accepts non-negative values"""  )
   return float(area_of_base * height  )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Union[str, Any]:




   """simple docstring"""





   if height < 0 or area_of_base < 0:
      raise ValueError("""vol_pyramid() only accepts non-negative values"""  )
   return area_of_base * height / 3.0
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_  )   ->   int:




   """simple docstring"""





   if radius < 0:
      raise ValueError("""vol_sphere() only accepts non-negative values"""  )
   # Volume is 4/3 * pi * radius cubed
   return 4 / 3 * pi * pow(lowerCamelCase_			,					3  )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_  )   ->   int:




   """simple docstring"""





   if radius < 0:
      raise ValueError("""vol_hemisphere() only accepts non-negative values"""  )
   # Volume is radius cubed * pi * 2/3
   return pow(lowerCamelCase_			,					3  ) * pi * 2 / 3
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   int:




   """simple docstring"""





   if height < 0 or radius < 0:
      raise ValueError("""vol_circular_cylinder() only accepts non-negative values"""  )
   # Volume is radius squared * height * pi
   return pow(lowerCamelCase_			,					2  ) * height * pi
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Optional[int]:




   """simple docstring"""





   if inner_radius < 0 or outer_radius < 0 or height < 0:
      raise ValueError(
          """vol_hollow_circular_cylinder() only accepts non-negative values"""  )
   if outer_radius <= inner_radius:
      raise ValueError("""outer_radius must be greater than inner_radius"""  )
   return pi * (pow(lowerCamelCase_			,					2  ) - pow(lowerCamelCase_			,					2  )) * height
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   Any:




   """simple docstring"""





   if radius_a < 0 or radius_a < 0 or height < 0:
      raise ValueError("""vol_conical_frustum() only accepts non-negative values"""  )
   return (
       1
       / 3
       * pi
       * height
       * (pow(lowerCamelCase_			,					2  ) + pow(lowerCamelCase_			,					2  ) + radius_a * radius_a)
   )
def       lowerCAmelCase_		( SCREAMING_SNAKE_CASE_			,					SCREAMING_SNAKE_CASE_  )   ->   List[str]:




   """simple docstring"""





   if torus_radius < 0 or tube_radius < 0:
      raise ValueError("""vol_torus() only accepts non-negative values"""  )
   return 2 * pow(lowerCamelCase_			,					2  ) * torus_radius * pow(lowerCamelCase_			,					2  )
def       lowerCAmelCase_		( )   ->   Any:




   """simple docstring"""
   print("""Volumes:"""  )
   print(F"Cube: {vol_cube(2  ) = }"  )  # = 8
   print(F"Cuboid: {vol_cuboid(2			,					2			,					2  ) = }"  )  # = 8
   print(F"Cone: {vol_cone(2			,					2  ) = }"  )  # ~= 1.33
   print(F"Right Circular Cone: {vol_right_circ_cone(2			,					2  ) = }"  )  # ~= 8.38
   print(F"Prism: {vol_prism(2			,					2  ) = }"  )  # = 4
   print(F"Pyramid: {vol_pyramid(2			,					2  ) = }"  )  # ~= 1.33
   print(F"Sphere: {vol_sphere(2  ) = }"  )  # ~= 33.5
   print(F"Hemisphere: {vol_hemisphere(2  ) = }"  )  # ~= 16.75
   print(F"Circular Cylinder: {vol_circular_cylinder(2			,					2  ) = }"  )  # ~= 25.1
   print(F"Torus: {vol_torus(2			,					2  ) = }"  )  # ~= 157.9
   print(F"Conical Frustum: {vol_conical_frustum(2			,					2			,					4  ) = }"  )  # ~= 58.6
   print(F"Spherical cap: {vol_spherical_cap(1			,					2  ) = }"  )  # ~= 5.24
   print(F"Spheres intersetion: {vol_spheres_intersect(2			,					2			,					1  ) = }"  )  # ~= 21.21
   print(F"Spheres union: {vol_spheres_union(2			,					2			,					1  ) = }"  )  # ~= 45.81
   print(
       F"Hollow Circular Cylinder: {vol_hollow_circular_cylinder(1			,					2			,					3  ) = }"  )  # ~= 28.3


if __name__ == "__main__":
     main()







'''simple docstring'''


from __future__ import annotations

from collections import deque
from enum import Enum
from math import atana, degrees
from sys import maxsize






class      __SCREAMING_SNAKE_CASE						(       lowerCamelCase_       ):







  '''simple docstring'''






  lowerCamelCase_		:Union[str, Any]						    =		1
  lowerCamelCase_		:List[Any]						    =		2
  lowerCamelCase_		:Optional[Any]						    =		3


  def __repr__( self						):




       '''simple docstring'''

       return F'''{self.__class__.__name__}.{self.name}'''







def 					_lowerCamelCase    (      lowerCamelCase_					:							tuple[int, int]	,     lowerCamelCase_					:							int	,     lowerCamelCase_					:							int   ):



     """simple docstring"""





     UpperCAmelCase_   ,			UpperCAmelCase_							:   Any   =      point
     return degrees(atana(y - miny	,     x - minx   )   )


def 					_lowerCamelCase    (      lowerCamelCase_					:							tuple[int, int]	,     lowerCamelCase_					:							tuple[int, int]	,     lowerCamelCase_					:							tuple[int, int]   ):



     """simple docstring"""





     UpperCAmelCase_   ,			UpperCAmelCase_							:   str   =      starting
     UpperCAmelCase_   ,			UpperCAmelCase_							:   Optional[int]   =      via
     UpperCAmelCase_   ,			UpperCAmelCase_							:   Dict   =      target
     UpperCAmelCase_							:   Optional[Any]   =      degrees(atana(ya - ya	,     xa - xa   )   )
     via_angle %= 360
     UpperCAmelCase_							:   Optional[Any]   =      degrees(atana(ya - ya	,     xa - xa   )   )
     target_angle %= 360
     # t-
     #  \ \
     #   \ v
     #    \|
     #     s
     # via_angle is always lower than target_angle, if direction is left.
     # If they are same, it means they are on a same line of convex hull.
     if target_angle > via_angle:
          return Direction.left
     elif target_angle == via_angle:
          return Direction.straight
     else:
          return Direction.right





def 					_lowerCamelCase    (      lowerCamelCase_					:							list[tuple[int, int]]   ):



     """simple docstring"""





     if len(lowerCamelCase_   ) <= 2:
          # There is no convex hull
          raise ValueError('graham_scan: argument must contain more than 3 points.'   )
     if len(lowerCamelCase_   ) == 3:
          return points
     # find the lowest and the most left point
     UpperCAmelCase_							:   Dict   =      0
     UpperCAmelCase_   ,			UpperCAmelCase_							:   Union[str, Any]   =      maxsize, maxsize
     for i, point in enumerate(lowerCamelCase_   ):
          UpperCAmelCase_							:   Union[str, Any]   =      point[0]
          UpperCAmelCase_							:   List[Any]   =      point[1]
          if y < miny:
               UpperCAmelCase_							:   Dict   =      y
               UpperCAmelCase_							:   Optional[int]   =      x
               UpperCAmelCase_							:   Optional[Any]   =      i
          if y == miny and x < minx:
               UpperCAmelCase_							:   Union[str, Any]   =      x
               UpperCAmelCase_							:   Dict   =      i

    # remove the lowest and the most left point from points for preparing for sort
     points.pop(lowerCamelCase_   )

     UpperCAmelCase_							:   List[str]   =      sorted(lowerCamelCase_	,     key=lambda lowerCamelCase_   : angle_comparer(lowerCamelCase_	,     lowerCamelCase_	,     lowerCamelCase_   )   )
     # This insert actually costs complexity,
     # and you should instead add (minx, miny) into stack later.
     # I'm using insert just for easy understanding.
     sorted_points.insert(0	,     (minx, miny)   )

     UpperCAmelCase_							:   deque[tuple[int, int]]   =      deque()
     stack.append(sorted_points[0]   )
     stack.append(sorted_points[1]   )
     stack.append(sorted_points[2]   )
     # In any ways, the first 3 points line are towards left.
     # Because we sort them the angle from minx, miny.
     UpperCAmelCase_							:   List[Any]   =      Direction.left

     for i in range(3	,     len(lowerCamelCase_   )   ):
          while True:
               UpperCAmelCase_							:   Optional[Any]   =      stack[-2]
               UpperCAmelCase_							:   Optional[int]   =      stack[-1]
               UpperCAmelCase_							:   str   =      sorted_points[i]
               UpperCAmelCase_							:   Union[str, Any]   =      check_direction(lowerCamelCase_	,     lowerCamelCase_	,     lowerCamelCase_   )

               if next_direction == Direction.left:
                    UpperCAmelCase_							:   Union[str, Any]   =      Direction.left
                    break
               if next_direction == Direction.straight:
                    if current_direction == Direction.left:
                         # We keep current_direction as left.
                         # Because if the straight line keeps as straight,
                         # we want to know if this straight line is towards left.
                         break
                    elif current_direction == Direction.right:
                         # If the straight line is towards right,
                         # every previous points on those straigh line is not convex hull.
                         stack.pop()
               if next_direction == Direction.right:
                    stack.pop()
          stack.append(sorted_points[i]   )
     return list(lowerCamelCase_   )






