





from __future__ import annotations

from collections.abc import Iterable

class 		__a							:



       def __init__(     self	:   List[str]					,			UpperCAmelCase	:   int					,			UpperCAmelCase	:   Optional[Any]			):
           lowerCAmelCase_		,		lowerCAmelCase_					:	List[Any]						=   float(UpperCAmelCase			), float(UpperCAmelCase			)



       def __eq__(     self	:   Dict					,			UpperCAmelCase	:   Tuple			):
           return self.x == other.x and self.y == other.y



       def __ne__(     self	:   str					,			UpperCAmelCase	:   str			):
           return not self == other



       def __gt__(     self	:   Optional[Any]					,			UpperCAmelCase	:   Optional[int]			):
           if self.x > other.x:
               return True
           elif self.x == other.x:
               return self.y > other.y
           return False



       def __lt__(     self	:   List[Any]					,			UpperCAmelCase	:   Union[str, Any]			):
           return not self > other



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



       def __le__(     self	:   Optional[Any]					,			UpperCAmelCase	:   Any			):
           if self.x < other.x:
               return True
           elif self.x == other.x:
               return self.y <= other.y
           return False



       def __repr__(     self	:   Union[str, Any]			):
           return F'({self.x}, {self.y})'



       def __hash__(     self	:   List[Any]			):
           return hash(self.x			)






def 			__UpperCamelCase				(			lowercase__	:       list[Point] | list[list[float]] | Iterable[list[float]]		,    )	->      list[Point]:
    '''simple docstring'''





    lowerCAmelCase_					:	list[Point]						=   []
    if list_of_tuples:
        for p in list_of_tuples:
            if isinstance(lowercase__		,    lowercase__		):
                points.append(lowercase__		)
            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				(			lowercase__	:       list[Point] | list[list[float]]		)	->      list[Point]:
    '''simple docstring'''





    if not hasattr(lowercase__		,    """__iter__"""		):
        lowerCAmelCase_					:	Optional[int]						=   f'Expecting an iterable object but got an non-iterable type {points}'
        raise ValueError(lowercase__		)

    if not points:
        lowerCAmelCase_					:	Optional[int]						=   f'Expecting a list of points but got {points}'
        raise ValueError(lowercase__		)

    return _construct_points(lowercase__		)
def 			__UpperCamelCase				(			lowercase__	:       Point		,    lowercase__	:       Point		,    lowercase__	:       Point		)	->      float:
    '''simple docstring'''





    lowerCAmelCase_					:	Dict						=   (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				(			lowercase__	:       list[Point]		)	->      list[Point]:
    '''simple docstring'''





    lowerCAmelCase_					:	Any						=   sorted(_validate_input(lowercase__		)		)
    lowerCAmelCase_					:	Optional[Any]						=   len(lowercase__		)
    lowerCAmelCase_					:	Any						=   set()

    for i in range(n - 1		):
        for j in range(i + 1		,    lowercase__		):
            lowerCAmelCase_					:	str						=   False
            lowerCAmelCase_					:	List[str]						=   True
            for k in range(lowercase__		):
                if k != i and k != j:
                    lowerCAmelCase_					:	List[str]						=   _det(points[i]		,    points[j]		,    points[k]		)

                    if det_k > 0:
                        lowerCAmelCase_					:	List[Any]						=   True
                    elif det_k < 0:
                        lowerCAmelCase_					:	Dict						=   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]:
                            lowerCAmelCase_					:	int						=   False
                            break

                if points_left_of_ij and points_right_of_ij:
                    lowerCAmelCase_					:	List[str]						=   False
                    break

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

    return sorted(lowercase__		)
def 			__UpperCamelCase				(			lowercase__	:       list[Point]		)	->      list[Point]:
    '''simple docstring'''





    lowerCAmelCase_					:	Optional[Any]						=   sorted(_validate_input(lowercase__		)		)
    lowerCAmelCase_					:	Optional[int]						=   len(lowercase__		)

    # 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

    lowerCAmelCase_					:	Dict						=   points[0]
    lowerCAmelCase_					:	Optional[int]						=   points[n - 1]

    lowerCAmelCase_					:	int						=   {left_most_point, right_most_point}
    lowerCAmelCase_					:	Optional[int]						=   []
    lowerCAmelCase_					:	Any						=   []

    for i in range(1		,    n - 1		):
        lowerCAmelCase_					:	Optional[int]						=   _det(lowercase__		,    lowercase__		,    points[i]		)

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

    _construct_hull(lowercase__		,    lowercase__		,    lowercase__		,    lowercase__		)
    _construct_hull(lowercase__		,    lowercase__		,    lowercase__		,    lowercase__		)

    return sorted(lowercase__		)
def 			__UpperCamelCase				(			lowercase__	:       list[Point]		,    lowercase__	:       Point		,    lowercase__	:       Point		,    lowercase__	:       set[Point]		)	->      None:
    '''simple docstring'''





    if points:
        lowerCAmelCase_					:	Tuple						=   None
        lowerCAmelCase_					:	str						=   float("""-inf"""		)
        lowerCAmelCase_					:	Tuple						=   []

        for p in points:
            lowerCAmelCase_					:	Dict						=   _det(lowercase__		,    lowercase__		,    lowercase__		)

            if det > 0:
                candidate_points.append(lowercase__		)

                if det > extreme_point_distance:
                    lowerCAmelCase_					:	str						=   det
                    lowerCAmelCase_					:	Tuple						=   p

        if extreme_point:
            _construct_hull(lowercase__		,    lowercase__		,    lowercase__		,    lowercase__		)
            convex_set.add(lowercase__		)
            _construct_hull(lowercase__		,    lowercase__		,    lowercase__		,    lowercase__		)
def 			__UpperCamelCase				(			lowercase__	:       list[Point]		)	->      list[Point]:
    '''simple docstring'''





    lowerCAmelCase_					:	Union[str, Any]						=   sorted(_validate_input(lowercase__		)		)
    lowerCAmelCase_					:	Optional[Any]						=   len(lowercase__		)

    lowerCAmelCase_					:	int						=   points[:2]
    for i in range(2		,    lowercase__		):
        lowerCAmelCase_					:	List[Any]						=   _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:
            lowerCAmelCase_					:	Union[str, Any]						=   points[i]
    i += 1

    for j in range(lowercase__		,    lowercase__		):
        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(lowercase__		) > 3 else convex_hull		)
def 			__UpperCamelCase				(			)	->      str:
    '''simple docstring'''





    lowerCAmelCase_					:	Tuple						=   [
        (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)]
    lowerCAmelCase_					:	Optional[int]						=   convex_hull_bf(lowercase__		)

    lowerCAmelCase_					:	Optional[int]						=   convex_hull_recursive(lowercase__		)
    assert results_bf == results_recursive

    lowerCAmelCase_					:	str						=   convex_hull_melkman(lowercase__		)
    assert results_bf == results_melkman

    print(lowercase__		)


if __name__ == "__main__":
  main()









def 			__UpperCamelCase				(			lowercase__	:       set		,    lowercase__	:       set		)	->      set:
    '''simple docstring'''





    assert isinstance(lowercase__		,    lowercase__		), f'The input value of [set_a={set_a}] is not a set'
    assert isinstance(lowercase__		,    lowercase__		), f'The input value of [set_b={set_b}] is not a set'

    return {a + b for a in set_a for b in set_b}


if __name__ == "__main__":
  from doctest import testmod

  testmod()


