
from __future__ import annotations

from scipy.special import comb  # type: ignore







class        lowerCAmelCase__      :

						def __init__(			self       ,							a				)					->      Dict:


								'''simple docstring'''

								_UpperCamelCase						    =	list_of_points
								# Degree determines the flexibility of the curve.
								# Degree = 1 will produce a straight line.
								_UpperCamelCase						    =	len(a				) - 1

						def  A_				(			self       ,							a				)					->      list[float]:


								'''simple docstring'''

								assert 0 <= t <= 1, "Time t must be between 0 and 1."
								_UpperCamelCase						    =	[]
								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  A_				(			self       ,							a				)					->      tuple[float, float]:


								'''simple docstring'''

								assert 0 <= t <= 1, "Time t must be between 0 and 1."

								_UpperCamelCase						    =	self.basis_function(a				)
								_UpperCamelCase						    =	0.0
								_UpperCamelCase						    =	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  A_				(			self       ,							a = 0.01				)					->      Any:


								'''simple docstring'''

								from matplotlib import pyplot as plt  # type: ignore

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

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

								_UpperCamelCase						    =	[i[0] for i in self.list_of_points]
								_UpperCamelCase						    =	[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





from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403


def 							__A(lowerCAmelCase						,  lowerCAmelCase						,  lowerCAmelCase = "x"						,  lowerCAmelCase = 1_0**-1_0						,  lowerCAmelCase = 1						,  )		->				complex:

		"""simple docstring"""






		_UpperCamelCase						    =	symbols(lowerCAmelCase )
		_UpperCamelCase						    =	lambdify(lowerCAmelCase						,  lowerCAmelCase )
		_UpperCamelCase						    =	lambdify(lowerCAmelCase						,  diff(lowerCAmelCase						,  lowerCAmelCase ) )

		_UpperCamelCase						    =	starting_point

		while True:
				if diff_function(lowerCAmelCase ) != 0:
						_UpperCamelCase						    =	prev_guess - multiplicity * func(lowerCAmelCase ) / diff_function(
						    lowerCAmelCase )
				else:
						raise ZeroDivisionError("""Could not find root""" ) from None

				# Precision is checked by comparing the difference of consecutive guesses
				if abs(next_guess - prev_guess ) < precision:
						return next_guess

				_UpperCamelCase						    =	next_guess


# Let's Execute
if __name__ == "__main__":
		# Find root of trigonometric function
		# Find value of pi
		print(F"""The root of sin(x) = 0 is {newton_raphson("sin(x)", 2)}""")

		# Find root of polynomial
		# Find fourth Root of 5
		print(F"""The root of x**4 - 5 = 0 is {newton_raphson("x**4 -5", 0.4 +5J)}""")

		# Find value of e
		print(
		    "The root of log(y) - 1 = 0 is ",
		    F"""{newton_raphson("log(y) - 1", 2, variable="y")}""",
		)

		# Exponential Roots
		print(
		    "The root of exp(x) - 1 = 0 is",
		    F"""{newton_raphson("exp(x) - 1", 10, precision=0.0_05)}""",
		)

		# Find root of cos(x)
		print(F"""The root of cos(x) = 0 is {newton_raphson("cos(x)", 0)}""")



