
from __future__ import annotations

from scipy.special import comb  # type: ignore

class     _UpperCamelCase			:






			'''simple docstring'''

			def __init__(    self	:    Dict  ,		snake_case_	:    list[tuple[float, float]] ):
							UpperCamelCase_:							List[str]		=  list_of_points
							# Degree determines the flexibility of the curve.
							# Degree = 1 will produce a straight line.
							UpperCamelCase_:							Tuple		=  len(snake_case_ ) - 1

			def  lowerCAmelCase__		(    self	:    Any  ,		snake_case_	:    float ):
							assert 0 <= t <= 1, "Time t must be between 0 and 1."
							UpperCamelCase_:							list[float]		=  []
							for i in range(len(self.list_of_points ) ):
											# basis function for each i
											output_values.append(
											    comb(self.degree  ,		snake_case_ ) * ((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(snake_case_ )  ,		5 ) == 1
							return output_values

			def  lowerCAmelCase__		(    self	:    List[str]  ,		snake_case_	:    float ):
							assert 0 <= t <= 1, "Time t must be between 0 and 1."

							UpperCamelCase_:							int		=  self.basis_function(snake_case_ )
							UpperCamelCase_:							int		=  0.0
							UpperCamelCase_:							List[Any]		=  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	:    str  ,		snake_case_	:    float = 0.01 ):
							from matplotlib import pyplot as plt  # type: ignore

							UpperCamelCase_:							list[float]		=  []  # x coordinates of points to plot
							UpperCamelCase_:							list[float]		=  []  # y coordinates of points to plot

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

							UpperCamelCase_:							List[Any]		=  [i[0] for i in self.list_of_points]
							UpperCamelCase_:							Tuple		=  [i[1] for i in self.list_of_points]

							plt.plot(
							    snake_case_  ,		snake_case_  ,		color="""blue"""  ,		label="""Curve of Degree """ + str(self.degree )  ,		)
							plt.scatter(snake_case_  ,		snake_case_  ,		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








def     A__   (  lowerCamelCase       ,      lowerCamelCase  )					->		list:
				UpperCamelCase_:							Optional[int]		=  word.split()

				def justify(lowerCamelCase       ,      lowerCamelCase       ,      lowerCamelCase  ) -> str:
								UpperCamelCase_:							Tuple		=  max_width - width
								UpperCamelCase_:							Optional[Any]		=  len(lowerCamelCase  )
								if len(lowerCamelCase  ) == 1:
												# if there is only word in line
												# just insert overall_spaces_count for the remainder of line
												return line[0] + " " * overall_spaces_count
								else:
												UpperCamelCase_:							List[Any]		=  words_count - 1
												# num_spaces_between_words_list[i] : tells you to insert
												# num_spaces_between_words_list[i] spaces
												# after word on line[i]
												UpperCamelCase_:							Optional[Any]		=  spaces_to_insert_between_words * [
												    overall_spaces_count // spaces_to_insert_between_words
												]
												UpperCamelCase_:							List[str]		=  (
												    overall_spaces_count % spaces_to_insert_between_words
												)
												# distribute spaces via round robin to the left words
												for i in range(lowerCamelCase  ):
																num_spaces_between_words_list[i] += 1
												UpperCamelCase_:							Dict		=  []
												for i in range(lowerCamelCase  ):
																# add the word
																aligned_words_list.append(line[i]  )
																# add the spaces to insert
																aligned_words_list.append(num_spaces_between_words_list[i] * """ """  )
												# just add the last word to the sentence
												aligned_words_list.append(line[-1]  )
												# join the aligned words list to form a justified line
												return "".join(lowerCamelCase  )

				UpperCamelCase_:							Optional[int]		=  []
				UpperCamelCase_:							list[str]		=  []
				UpperCamelCase_:							List[str]		=  0
				for word in words:
								if width + len(lowerCamelCase  ) + len(lowerCamelCase  ) <= max_width:
												# keep adding words until we can fill out max_width
												# width = sum of length of all words (without overall_spaces_count)
												# len(word) = length of current word
												# len(line) = number of overall_spaces_count to insert between words
												line.append(lowerCamelCase  )
												width += len(lowerCamelCase  )
								else:
												# justify the line and add it to result
												answer.append(justify(lowerCamelCase       ,      lowerCamelCase       ,      lowerCamelCase  )  )
												# reset new line and new width
												UpperCamelCase_,				UpperCamelCase_:							List[str]		=  [word], len(lowerCamelCase  )
				UpperCamelCase_:							List[str]		=  max_width - width - len(lowerCamelCase  )
				answer.append(""" """.join(lowerCamelCase  ) + (remaining_spaces + 1) * """ """  )
				return answer


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

							testmod()






