


from __future__ import annotations

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

# initial triangle of Koch snowflake
_lowercase							:   List[str]						=						numpy.array([0, 0])
_lowercase							:   Tuple						=						numpy.array([0.5, 0.8_66_02_54])
_lowercase							:   Optional[int]						=						numpy.array([1, 0])
_lowercase							:   List[str]						=						[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]





def 	_lowerCAmelCase					( UpperCamelCase__:  list[numpy.ndarray]				,       UpperCamelCase__:  int    )			-> list[numpy.ndarray]:





	"""simple docstring"""







	A             =					initial_vectors
	for _ in range(UpperCamelCase__    ):
		A             =					iteration_step(UpperCamelCase__    )
	return vectors





def 	_lowerCAmelCase					( UpperCamelCase__:  list[numpy.ndarray]    )			-> list[numpy.ndarray]:





	"""simple docstring"""







	A             =					[]
	for i, start_vector in enumerate(vectors[:-1]    ):
		A             =					vectors[i + 1]
		new_vectors.append(UpperCamelCase__    )
		A             =					end_vector - start_vector
		new_vectors.append(start_vector + difference_vector / 3    )
		new_vectors.append(
		    start_vector + difference_vector / 3 + rotate(difference_vector / 3				,       60    )    )
		new_vectors.append(start_vector + difference_vector * 2 / 3    )
	new_vectors.append(vectors[-1]    )
	return new_vectors





def 	_lowerCAmelCase					( UpperCamelCase__:  numpy.ndarray				,       UpperCamelCase__:  float    )			-> numpy.ndarray:





	"""simple docstring"""







	A             =					numpy.radians(UpperCamelCase__    )
	A       ,      A             =					numpy.cos(UpperCamelCase__    ), numpy.sin(UpperCamelCase__    )
	A             =					numpy.array(((c, -s), (s, c))    )
	return numpy.dot(UpperCamelCase__				,       UpperCamelCase__    )





def 	_lowerCAmelCase					( UpperCamelCase__:  list[numpy.ndarray]    )			-> None:





	"""simple docstring"""







	A             =					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()
	A       ,      A             =					zip(*UpperCamelCase__    )
	plt.plot(UpperCamelCase__				,       UpperCamelCase__    )
	plt.show()


if __name__ == "__main__":
				import doctest

				doctest.testmod()

				_lowercase							:   Optional[int]						=						iterate(INITIAL_VECTORS, 5)
				plot(processed_vectors)







def 	_lowerCAmelCase					( UpperCamelCase__:  str				,       UpperCamelCase__:  int				,       UpperCamelCase__:  Any=False    )			-> str:





	"""simple docstring"""







	if isinstance(UpperCamelCase__				,       UpperCamelCase__    ) and isinstance(UpperCamelCase__				,       UpperCamelCase__    ):
		A             =					len(set_a.intersection(UpperCamelCase__    )    )

		if alternative_union:
			A             =					len(UpperCamelCase__    ) + len(UpperCamelCase__    )
		else:
			A             =					len(set_a.union(UpperCamelCase__    )    )

		return intersection / union

	if isinstance(UpperCamelCase__				,       (list, tuple)    ) and isinstance(UpperCamelCase__				,       (list, tuple)    ):
		A             =					[element for element in set_a if element in set_b]

		if alternative_union:
			A             =					len(UpperCamelCase__    ) + len(UpperCamelCase__    )
			return len(UpperCamelCase__    ) / union
		else:
			A             =					set_a + [element for element in set_b if element not in set_a]
			return len(UpperCamelCase__    ) / len(UpperCamelCase__    )

		return len(UpperCamelCase__    ) / len(UpperCamelCase__    )
	return None


if __name__ == "__main__":
				_lowercase							:   List[Any]						=						{"a", "b", "c", "d", "e"}
				_lowercase							:   List[Any]						=						{"c", "d", "e", "f", "h", "i"}
				print(jaccard_similarity(set_a, set_b))



