'''simple docstring'''






def  UpperCamelCase__					(					_lowercase    :				int   ,    _lowercase    :				Any     )							->	int:
  return (pointa[0] - pointa[0]) ** 2 + (pointa[1] - pointa[1]) ** 2






def  UpperCamelCase__					(					_lowercase    :				Tuple   ,    _lowercase    :				List[Any]=0     )							->	Union[str, Any]:
  return sorted(__lowercase   ,    key=lambda _lowercase     : x[column]     )






def  UpperCamelCase__					(					_lowercase    :				Optional[Any]   ,    _lowercase    :				str   ,    _lowercase    :				Optional[Any]=float("""inf"""     )     )							->	str:




  for i in range(points_counts - 1     ):
    for j in range(i + 1   ,    __lowercase     ):
      __UpperCAmelCase:					int		= euclidean_distance_sqr(points[i]   ,    points[j]     )
      if current_dis < min_dis:
        __UpperCAmelCase:					int		= current_dis
  return min_dis






def  UpperCamelCase__					(					_lowercase    :				Tuple   ,    _lowercase    :				int   ,    _lowercase    :				Optional[int]=float("""inf"""     )     )							->	List[Any]:




  for i in range(min(6   ,    points_counts - 1     )   ,    __lowercase     ):
    for j in range(max(0   ,    i - 6     )   ,    __lowercase     ):
      __UpperCAmelCase:					Dict		= euclidean_distance_sqr(points[i]   ,    points[j]     )
      if current_dis < min_dis:
        __UpperCAmelCase:					Any		= current_dis
  return min_dis






def  UpperCamelCase__					(					_lowercase    :				int   ,    _lowercase    :				Dict   ,    _lowercase    :				Union[str, Any]     )							->	Union[str, Any]:




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

  # recursion
  __UpperCAmelCase:					Any		= points_counts // 2
  __UpperCAmelCase:					Dict		= closest_pair_of_points_sqr(
      __lowercase   ,    points_sorted_on_y[:mid]   ,    __lowercase     )
  __UpperCAmelCase:					str		= closest_pair_of_points_sqr(
      __lowercase   ,    points_sorted_on_y[mid:]   ,    points_counts - mid     )
  __UpperCAmelCase:					Optional[Any]		= min(__lowercase   ,    __lowercase     )
  __UpperCAmelCase:					str		= []
  for point in points_sorted_on_x:
    if abs(point[0] - points_sorted_on_x[mid][0]     ) < closest_pair_dis:
      cross_strip.append(__lowercase     )

  __UpperCAmelCase:					Tuple		= dis_between_closest_in_strip(
      __lowercase   ,    len(__lowercase     )   ,    __lowercase     )
  return min(__lowercase   ,    __lowercase     )






def  UpperCamelCase__					(					_lowercase    :				Any   ,    _lowercase    :				List[Any]     )							->	List[str]:
  __UpperCAmelCase:					Union[str, Any]		= column_based_sort(__lowercase   ,    column=0     )
  __UpperCAmelCase:					List[str]		= column_based_sort(__lowercase   ,    column=1     )
  return (
      closest_pair_of_points_sqr(
          __lowercase   ,    __lowercase   ,    __lowercase     )
  ) ** 0.5


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


'''simple docstring'''






from __future__ import annotations

import matplotlib.pyplot as plt  # type: ignore
import numpy

# initial triangle of Koch snowflake
SCREAMING_SNAKE_CASE_	   =			numpy.array([0, 0])
SCREAMING_SNAKE_CASE_	   =			numpy.array([0.5, 0.866_0254])
SCREAMING_SNAKE_CASE_	   =			numpy.array([1, 0])
SCREAMING_SNAKE_CASE_	   =			[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]






def  UpperCamelCase__					(					_lowercase    :				list[numpy.ndarray]   ,    _lowercase    :				int     )							->	list[numpy.ndarray]:
  __UpperCAmelCase:					int		= initial_vectors
  for _ in range(_lowercase     ):
    __UpperCAmelCase:					Any		= iteration_step(_lowercase     )
  return vectors






def  UpperCamelCase__					(					_lowercase    :				list[numpy.ndarray]     )							->	list[numpy.ndarray]:
  __UpperCAmelCase:					Optional[int]		= []
  for i, start_vector in enumerate(vectors[:-1]     ):
    __UpperCAmelCase:					Union[str, Any]		= vectors[i + 1]
    new_vectors.append(_lowercase     )
    __UpperCAmelCase:					Optional[int]		= end_vector - start_vector
    new_vectors.append(start_vector + difference_vector / 3     )
    new_vectors.append(
        start_vector + difference_vector / 3 + rotate(difference_vector / 3   ,    6_0     )     )
    new_vectors.append(start_vector + difference_vector * 2 / 3     )
  new_vectors.append(vectors[-1]     )
  return new_vectors






def  UpperCamelCase__					(					_lowercase    :				numpy.ndarray   ,    _lowercase    :				float     )							->	numpy.ndarray:
  __UpperCAmelCase:					Tuple		= numpy.radians(_lowercase     )
  __UpperCAmelCase, __UpperCAmelCase:					Optional[Any]		= numpy.cos(_lowercase     ), numpy.sin(_lowercase     )
  __UpperCAmelCase:					Tuple		= numpy.array(((c, -s), (s, c))     )
  return numpy.dot(_lowercase   ,    _lowercase     )






def  UpperCamelCase__					(					_lowercase    :				list[numpy.ndarray]     )							->	None:
  __UpperCAmelCase:					Union[str, Any]		= plt.gca()
  axes.set_aspect("""equal"""     )

  # matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
  # y-coordinates as inputs, which are constructed from the vector-list using
  # zip()
  __UpperCAmelCase, __UpperCAmelCase:					Dict		= zip(*_lowercase     )
  plt.plot(_lowercase   ,    _lowercase     )
  plt.show()


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    SCREAMING_SNAKE_CASE_	   =			iterate(INITIAL_VECTORS, 5)
    plot(processed_vectors)