



from __future__ import annotations

from scipy.special import comb  # type: ignore
class 				__A					:





			def __init__(self							,    __magic_name__				):
						lowerCamelCase__ :      Union[str, Any]								=		list_of_points
						# Degree determines the flexibility of the curve.
						# Degree = 1 will produce a straight line.
						lowerCamelCase__ :      Union[str, Any]								=		len(__magic_name__				) - 1





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





			def        _snake_case    (self							,    __magic_name__				):
						assert 0 <= t <= 1, "Time t must be between 0 and 1."

						lowerCamelCase__ :      List[Any]								=		self.basis_function(__magic_name__				)
						lowerCamelCase__ :      Union[str, Any]								=		0.0
						lowerCamelCase__ :      str								=		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        _snake_case    (self							,    __magic_name__ = 0.01				):
						from matplotlib import pyplot as plt  # type: ignore

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

						lowerCamelCase__ :      str								=		0.0
						while t <= 1:
									lowerCamelCase__ :      Optional[Any]								=		self.bezier_curve_function(__magic_name__				)
									to_plot_x.append(value[0]				)
									to_plot_y.append(value[1]				)
									t += step_size

						lowerCamelCase__ :      Optional[int]								=		[i[0] for i in self.list_of_points]
						lowerCamelCase__ :      Union[str, Any]								=		[i[1] for i in self.list_of_points]

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








from __future__ import annotations

import math
from collections import Counter
from string import ascii_lowercase






def      _A				(UpperCamelCase		:      str )  ->None:



			'''simple docstring'''







			lowerCamelCase__							,lowerCamelCase__ :      List[str]								=		analyze_text(UpperCamelCase )
			lowerCamelCase__ :      Any								=		list(""" """ + ascii_lowercase )
			# what is our total sum of probabilities.
			lowerCamelCase__ :      str								=		sum(single_char_strings.values() )

			# one length string
			lowerCamelCase__ :      Union[str, Any]								=		0
			# for each alpha we go in our dict and if it is in it we calculate entropy
			for ch in my_alphas:
						if ch in single_char_strings:
									lowerCamelCase__ :      List[Any]								=		single_char_strings[ch]
									lowerCamelCase__ :      List[str]								=		my_str / all_sum
									my_fir_sum += prob * math.loga(UpperCamelCase )  # entropy formula.

    # print entropy
			print(f"{round(-1 * my_fir_sum ):.1f}" )

			# two len string
			lowerCamelCase__ :      str								=		sum(two_char_strings.values() )
			lowerCamelCase__ :      int								=		0
			# for each alpha (two in size) calculate entropy.
			for cha in my_alphas:
						for cha in my_alphas:
									lowerCamelCase__ :      str								=		cha + cha
									if sequence in two_char_strings:
												lowerCamelCase__ :      str								=		two_char_strings[sequence]
												lowerCamelCase__ :      int								=		int(UpperCamelCase ) / all_sum
												my_sec_sum += prob * math.loga(UpperCamelCase )

    # print second entropy
			print(f"{round(-1 * my_sec_sum ):.1f}" )

			# print the difference between them
			print(f"{round((-1 * my_sec_sum) - (-1 * my_fir_sum) ):.1f}" )






def      _A				(UpperCamelCase		:      str )  ->tuple[dict, dict]:



			'''simple docstring'''







			lowerCamelCase__ :      Optional[int]								=		Counter()  # type: ignore
			lowerCamelCase__ :      List[Any]								=		Counter()  # type: ignore
			single_char_strings[text[-1]] += 1

			# first case when we have space at start.
			two_char_strings[" " + text[0]] += 1
			for i in range(0      ,			len(UpperCamelCase ) - 1 ):
						single_char_strings[text[i]] += 1
						two_char_strings[text[i : i + 2]] += 1
			return single_char_strings, two_char_strings






def      _A				()  ->List[str]:



			'''simple docstring'''







			import doctest

			doctest.testmod()
			# text = (
			#     "Had repulsive dashwoods suspicion sincerity but advantage now him. Remark "
			#     "easily garret nor nay. Civil those mrs enjoy shy fat merry. You greatest "
			#     "jointure saw horrible. He private he on be imagine suppose. Fertile "
			#     "beloved evident through no service elderly is. Blind there if every no so "
			#     "at. Own neglected you preferred way sincerity delivered his attempted. To "
			#     "of message cottage windows do besides against uncivil.  Delightful "
			#     "unreserved impossible few estimating men favourable see entreaties. She "
			#     "propriety immediate was improving. He or entrance humoured likewise "
			#     "moderate. Much nor game son say feel. Fat make met can must form into "
			#     "gate. Me we offending prevailed discovery. "
			# )

			# calculate_prob(text)


if __name__ == "__main__":
				main()



