




'''simple docstring'''


from __future__ import annotations

from scipy.special import comb  # type: ignore

class       A   :







       def __init__(  self	:  List[str]							,      lowerCAmelCase_	:  list[tuple[float, float]] )      ->							Optional[Any]:


              """simple docstring"""


              _a     					=    list_of_points
              # Degree determines the flexibility of the curve.
              # Degree = 1 will produce a straight line.
              _a     					=    len(lowerCAmelCase_ ) - 1







       def        __lowerCAmelCase      (  self	:  str							,      lowerCAmelCase_	:  float )      ->							list[float]:


              """simple docstring"""


              assert 0 <= t <= 1, "Time t must be between 0 and 1."
              _a     					=    []
              for i in range(len(self.list_of_points ) ):
                     # basis function for each i
                     output_values.append(
                         comb(self.degree							,      lowerCAmelCase_ ) * ((1 - t) ** (self.degree - i)) * (t**i) )
              # the basis must sum up to 1 for it to produce a valid Bezier curve.
              assert round(sum(lowerCAmelCase_ )							,      5 ) == 1
              return output_values







       def        __lowerCAmelCase      (  self	:  Optional[Any]							,      lowerCAmelCase_	:  float )      ->							tuple[float, float]:


              """simple docstring"""


              assert 0 <= t <= 1, "Time t must be between 0 and 1."

              _a     					=    self.basis_function(lowerCAmelCase_ )
              _a     					=    0.0
              _a     					=    0.0
              for i in range(len(self.list_of_points ) ):
                     # For all points, sum up the product of i-th basis function and i-th point.
                     x += basis_function[i] * self.list_of_points[i][0]
                     y += basis_function[i] * self.list_of_points[i][1]
              return (x, y)







       def        __lowerCAmelCase      (  self	:  Dict							,      lowerCAmelCase_	:  float = 0.0_1 )      ->							Union[str, Any]:


              """simple docstring"""


              from matplotlib import pyplot as plt  # type: ignore

              _a     					=    []  # x coordinates of points to plot
              _a     					=    []  # y coordinates of points to plot

              _a     					=    0.0
              while t <= 1:
                     _a     					=    self.bezier_curve_function(lowerCAmelCase_ )
                     to_plot_x.append(value[0] )
                     to_plot_y.append(value[1] )
                     t += step_size

              _a     					=    [i[0] for i in self.list_of_points]
              _a     					=    [i[1] for i in self.list_of_points]

              plt.plot(
                  lowerCAmelCase_							,      lowerCAmelCase_							,      color='''blue'''							,      label='''Curve of Degree ''' + str(self.degree )							,      )
              plt.scatter(lowerCAmelCase_							,      lowerCAmelCase_							,      color='''red'''							,      label='''Control Points''' )
              plt.legend()
              plt.show()


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
    BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
    BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3







'''simple docstring'''


def 		snake_case_   (UpperCamelCase	:  str			,						UpperCamelCase	:  Any				):





							'''simple docstring'''


							return (pointa[0] - pointa[0]) ** 2 + (pointa[1] - pointa[1]) ** 2
def 		snake_case_   (UpperCamelCase	:  Any			,						UpperCamelCase	:  str=0				):





							'''simple docstring'''


							return sorted(UpperCamelCase			,						key=lambda UpperCamelCase				: x[column]				)
def 		snake_case_   (UpperCamelCase	:  Optional[int]			,						UpperCamelCase	:  Any			,						UpperCamelCase	:  Union[str, Any]=float('''inf'''				)				):





							'''simple docstring'''


							for i in range(points_counts - 1				):
														for j in range(i + 1			,						UpperCamelCase				):
																					_a     					=    euclidean_distance_sqr(points[i]			,						points[j]				)
																					if current_dis < min_dis:
																												_a     					=    current_dis
							return min_dis
def 		snake_case_   (UpperCamelCase	:  int			,						UpperCamelCase	:  Tuple			,						UpperCamelCase	:  List[str]=float('''inf'''				)				):





							'''simple docstring'''


							for i in range(min(6			,						points_counts - 1				)			,						UpperCamelCase				):
														for j in range(max(0			,						i - 6				)			,						UpperCamelCase				):
																					_a     					=    euclidean_distance_sqr(points[i]			,						points[j]				)
																					if current_dis < min_dis:
																												_a     					=    current_dis
							return min_dis
def 		snake_case_   (UpperCamelCase	:  int			,						UpperCamelCase	:  List[Any]			,						UpperCamelCase	:  int				):





							'''simple docstring'''


							if points_counts <= 3:
														return dis_between_closest_pair(UpperCamelCase			,						UpperCamelCase				)

							# recursion
							_a     					=    points_counts // 2
							_a     					=    closest_pair_of_points_sqr(
							    UpperCamelCase			,						points_sorted_on_y[:mid]			,						UpperCamelCase				)
							_a     					=    closest_pair_of_points_sqr(
							    UpperCamelCase			,						points_sorted_on_y[mid:]			,						points_counts - mid				)
							_a     					=    min(UpperCamelCase			,						UpperCamelCase				)
							_a     					=    []
							for point in points_sorted_on_x:
														if abs(point[0] - points_sorted_on_x[mid][0]				) < closest_pair_dis:
																					cross_strip.append(UpperCamelCase				)

							_a     					=    dis_between_closest_in_strip(
							    UpperCamelCase			,						len(UpperCamelCase				)			,						UpperCamelCase				)
							return min(UpperCamelCase			,						UpperCamelCase				)
def 		snake_case_   (UpperCamelCase	:  Optional[int]			,						UpperCamelCase	:  List[str]				):





							'''simple docstring'''


							_a     					=    column_based_sort(UpperCamelCase			,						column=0				)
							_a     					=    column_based_sort(UpperCamelCase			,						column=1				)
							return (
							    closest_pair_of_points_sqr(
							        UpperCamelCase			,						UpperCamelCase			,						UpperCamelCase				)
							) ** 0.5


if __name__ == "__main__":
				_snake_case				:				int       							=		[(2, 3), (12, 30), (40, 50), (5, 1), (12, 10), (3, 4)]
				print('Distance:', closest_pair_of_points(points, len(points)))

