
'''simple docstring'''
from __future__ import annotations

from collections.abc import Iterable






class 				UpperCAmelCase_			:





 def __init__(      self	,    lowercase_	,    lowercase_):
     snake_case_	:      Optional[int]            =  float(SCREAMING_SNAKE_CASE__), float(SCREAMING_SNAKE_CASE__)





 def __eq__(      self	,    lowercase_):
     return self.x == other.x and self.y == other.y





 def __ne__(      self	,    lowercase_):
     return not self == other





 def __gt__(      self	,    lowercase_):
     if self.x > other.x:
         return True
     elif self.x == other.x:
         return self.y > other.y
     return False





 def __lt__(      self	,    lowercase_):
     return not self > other





 def __ge__(      self	,    lowercase_):
     if self.x > other.x:
         return True
     elif self.x == other.x:
         return self.y >= other.y
     return False





 def __le__(      self	,    lowercase_):
     if self.x < other.x:
         return True
     elif self.x == other.x:
         return self.y <= other.y
     return False





 def __repr__(      self):
     return F'({self.x}, {self.y})'





 def __hash__(      self):
     return hash(self.x)


def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE,						):
    """simple docstring"""






    snake_case_	:      list[Point]            =  []
    if list_of_tuples:
        for p in list_of_tuples:
            if isinstance(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    ):
                points.append(__SCREAMING_SNAKE_CASE    )
            else:
                try:
                    points.append(Point(p[0],						p[1]    )    )
                except (IndexError, TypeError):
                    print(
                        f'Ignoring deformed point {p}. All points'
                        " must have at least 2 coordinates."    )
    return points

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    if not hasattr(__SCREAMING_SNAKE_CASE,						"__iter__"    ):
        snake_case_	:      List[Any]            =  f'Expecting an iterable object but got an non-iterable type {points}'
        raise ValueError(__SCREAMING_SNAKE_CASE    )

    if not points:
        snake_case_	:      Tuple            =  f'Expecting a list of points but got {points}'
        raise ValueError(__SCREAMING_SNAKE_CASE    )

    return _construct_points(__SCREAMING_SNAKE_CASE    )

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    snake_case_	:      Optional[int]            =  (a.x * b.y + b.x * c.y + c.x * a.y) - (a.y * b.x + b.y * c.x + c.y * a.x)
    return det

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    snake_case_	:      int            =  sorted(_validate_input(__SCREAMING_SNAKE_CASE    )    )
    snake_case_	:      int            =  len(__SCREAMING_SNAKE_CASE    )
    snake_case_	:      Union[str, Any]            =  set()

    for i in range(n - 1    ):
        for j in range(i + 1,						__SCREAMING_SNAKE_CASE    ):
            snake_case_	:      List[str]            =  False
            snake_case_	:      Optional[Any]            =  True
            for k in range(__SCREAMING_SNAKE_CASE    ):
                if k != i and k != j:
                    snake_case_	:      Union[str, Any]            =  _det(points[i],						points[j],						points[k]    )

                    if det_k > 0:
                        snake_case_	:      Union[str, Any]            =  True
                    elif det_k < 0:
                        snake_case_	:      int            =  True
                    else:
                        # point[i], point[j], point[k] all lie on a straight line
                        # if point[k] is to the left of point[i] or it's to the
                        # right of point[j], then point[i], point[j] cannot be
                        # part of the convex hull of A
                        if points[k] < points[i] or points[k] > points[j]:
                            snake_case_	:      Dict            =  False
                            break

                if points_left_of_ij and points_right_of_ij:
                    snake_case_	:      str            =  False
                    break

            if ij_part_of_convex_hull:
                convex_set.update([points[i], points[j]]    )

    return sorted(__SCREAMING_SNAKE_CASE    )

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    snake_case_	:      List[Any]            =  sorted(_validate_input(__SCREAMING_SNAKE_CASE    )    )
    snake_case_	:      Any            =  len(__SCREAMING_SNAKE_CASE    )

    # divide all the points into an upper hull and a lower hull
    # the left most point and the right most point are definitely
    # members of the convex hull by definition.
    # use these two anchors to divide all the points into two hulls,
    # an upper hull and a lower hull.

    # all points to the left (above) the line joining the extreme points belong to the
    # upper hull
    # all points to the right (below) the line joining the extreme points below to the
    # lower hull
    # ignore all points on the line joining the extreme points since they cannot be
    # part of the convex hull

    snake_case_	:      Dict            =  points[0]
    snake_case_	:      int            =  points[n - 1]

    snake_case_	:      Tuple            =  {left_most_point, right_most_point}
    snake_case_	:      Optional[Any]            =  []
    snake_case_	:      Optional[Any]            =  []

    for i in range(1,						n - 1    ):
        snake_case_	:      Dict            =  _det(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						points[i]    )

        if det > 0:
            upper_hull.append(points[i]    )
        elif det < 0:
            lower_hull.append(points[i]    )

    _construct_hull(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    )
    _construct_hull(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    )

    return sorted(__SCREAMING_SNAKE_CASE    )

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    if points:
        snake_case_	:      int            =  None
        snake_case_	:      Any            =  float("-inf"    )
        snake_case_	:      Union[str, Any]            =  []

        for p in points:
            snake_case_	:      List[str]            =  _det(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    )

            if det > 0:
                candidate_points.append(__SCREAMING_SNAKE_CASE    )

                if det > extreme_point_distance:
                    snake_case_	:      Tuple            =  det
                    snake_case_	:      List[str]            =  p

        if extreme_point:
            _construct_hull(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    )
            convex_set.add(__SCREAMING_SNAKE_CASE    )
            _construct_hull(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    )

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
    """simple docstring"""






    snake_case_	:      Union[str, Any]            =  sorted(_validate_input(__SCREAMING_SNAKE_CASE    )    )
    snake_case_	:      Optional[Any]            =  len(__SCREAMING_SNAKE_CASE    )

    snake_case_	:      Optional[Any]            =  points[:2]
    for i in range(2,						__SCREAMING_SNAKE_CASE    ):
        snake_case_	:      List[str]            =  _det(convex_hull[1],						convex_hull[0],						points[i]    )
        if det > 0:
            convex_hull.insert(0,						points[i]    )
            break
        elif det < 0:
            convex_hull.append(points[i]    )
            break
        else:
            snake_case_	:      Any            =  points[i]
    i += 1

    for j in range(__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE    ):
        if (
            _det(convex_hull[0],						convex_hull[-1],						points[j]    ) > 0
            and _det(convex_hull[-1],						convex_hull[0],						points[1]    ) < 0
        ):
            # The point lies within the convex hull
            continue

        convex_hull.insert(0,						points[j]    )
        convex_hull.append(points[j]    )
        while _det(convex_hull[0],						convex_hull[1],						convex_hull[2]    ) >= 0:
            del convex_hull[1]
        while _det(convex_hull[-1],						convex_hull[-2],						convex_hull[-3]    ) <= 0:
            del convex_hull[-2]

        # `convex_hull` is contains the convex hull in circular order
    return sorted(convex_hull[1:] if len(__SCREAMING_SNAKE_CASE    ) > 3 else convex_hull    )

def   UpperCamelCase_       (		):
    """simple docstring"""






    snake_case_	:      str            =  [
        (0, 3),
        (2, 2),
        (1, 1),
        (2, 1),
        (3, 0),
        (0, 0),
        (3, 3),
        (2, -1),
        (2, -4),
        (1, -3),
    ]
    # the convex set of points is
    # [(0, 0), (0, 3), (1, -3), (2, -4), (3, 0), (3, 3)]
    snake_case_	:      Dict            =  convex_hull_bf(__SCREAMING_SNAKE_CASE    )

    snake_case_	:      str            =  convex_hull_recursive(__SCREAMING_SNAKE_CASE    )
    assert results_bf == results_recursive

    snake_case_	:      List[Any]            =  convex_hull_melkman(__SCREAMING_SNAKE_CASE    )
    assert results_bf == results_melkman

    print(__SCREAMING_SNAKE_CASE    )


if __name__ == "__main__":
  main()






'''simple docstring'''
from __future__ import annotations

from collections.abc import Callable

def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE,						__SCREAMING_SNAKE_CASE = 1_0_0,						):
    """simple docstring"""






    snake_case_	:      Optional[Any]            =  x_start
    snake_case_	:      str            =  fnc(__SCREAMING_SNAKE_CASE    )
    snake_case_	:      List[str]            =  0.0

    for _ in range(__SCREAMING_SNAKE_CASE    ):
        # Approximates small segments of curve as linear and solve
        # for trapezoidal area
        snake_case_	:      List[str]            =  (x_end - x_start) / steps + xa
        snake_case_	:      Optional[int]            =  fnc(__SCREAMING_SNAKE_CASE    )
        area += abs(fxa + fxa    ) * (xa - xa) / 2

        # Increment step
        snake_case_	:      List[Any]            =  xa
        snake_case_	:      Optional[Any]            =  fxa
    return area


if __name__ == "__main__":

  def   UpperCamelCase_       (		__SCREAMING_SNAKE_CASE    ):
      """simple docstring"""






      return x**3

  print("f(x) = x^3")
  print("The area between the curve, x = -10, x = 10 and the x axis is:")
  a_     =  10
  while i <= 10_0000:
    a_     =  trapezoidal_area(f, -5, 5, i)
    print(f"""with {i} steps: {area}""")
    i *= 10




