





from __future__ import annotations

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




					def __init__(						self			,  lowerCAmelCase_					)						->							int:
										_snake_case													=			list_of_points
										# Degree determines the flexibility of the curve.
										# Degree = 1 will produce a straight line.
										_snake_case													=			len(_A					) - 1




					def 		lowerCAmelCase     (						self			,  lowerCAmelCase_					)						->							List[str]:
										assert 0 <= t <= 1, "Time t must be between 0 and 1."
										_snake_case													=			[]
										for i in range(len(self.list_of_points					)					):
															# basis function for each i
															output_values.append(
															    comb(self.degree			,  _A					) * ((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(_A					)			,  5					) == 1
										return output_values




					def 		lowerCAmelCase     (						self			,  lowerCAmelCase_					)						->							List[Any]:
										assert 0 <= t <= 1, "Time t must be between 0 and 1."

										_snake_case													=			self.basis_function(_A					)
										_snake_case													=			0.0
										_snake_case													=			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			,  lowerCAmelCase_ = 0.01					)						->							Tuple:
										from matplotlib import pyplot as plt  # type: ignore

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

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

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

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








import random





def   lowerCamelCase__       (       UpperCamelCase__     :    int    ,      UpperCamelCase__     :    float    ,      UpperCamelCase__     :    bool = False  )     ->	dict:



					'''simple docstring'''




					_snake_case													=			{i: [] for i in range(UpperCamelCase__  )}

					# if probability is greater or equal than 1, then generate a complete graph
					if probability >= 1:
										return complete_graph(UpperCamelCase__  )
					# if probability is lower or equal than 0, then return a graph without edges
					if probability <= 0:
										return graph

					# for each couple of nodes, add an edge from u to v
					# if the number randomly generated is greater than probability probability
					for i in range(UpperCamelCase__  ):
										for j in range(i + 1    ,      UpperCamelCase__  ):
															if random.random() < probability:
																				graph[i].append(UpperCamelCase__  )
																				if not directed:
																									# if the graph is undirected, add an edge in from j to i, either
																									graph[j].append(UpperCamelCase__  )
					return graph





def   lowerCamelCase__       (       UpperCamelCase__     :    int  )     ->	dict:



					'''simple docstring'''




					return {
					    i: [j for j in range(UpperCamelCase__  ) if i != j] for i in range(UpperCamelCase__  )
					}


if __name__ == "__main__":
					import doctest

					doctest.testmod()



