"""simple docstring"""
from __future__ import annotations

from scipy.special import comb  # type: ignore




class    __snake_case       :



      def __init__(						self    ,			lowercase)	->			int:





        '''simple docstring'''





        a__:     Dict										=	list_of_points
        # Degree determines the flexibility of the curve.
        # Degree = 1 will produce a straight line.
        a__:     List[Any]										=	len(lowercase) - 1



      def   lowerCamelCase_   (						self    ,			lowercase)	->			list[float]:





        '''simple docstring'''





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



      def   lowerCamelCase_   (						self    ,			lowercase)	->			tuple[float, float]:





        '''simple docstring'''





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

        a__:     Tuple										=	self.basis_function(lowercase)
        a__:     List[str]										=	0.0
        a__:     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    ,			lowercase = 0.01)	->			str:





        '''simple docstring'''





        from matplotlib import pyplot as plt  # type: ignore

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

        a__:     List[str]										=	0.0
        while t <= 1:
          a__:     Union[str, Any]										=	self.bezier_curve_function(lowercase)
          to_plot_x.append(value[0])
          to_plot_y.append(value[1])
          t += step_size

        a__:     Dict										=	[i[0] for i in self.list_of_points]
        a__:     Optional[Any]										=	[i[1] for i in self.list_of_points]

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



"""simple docstring"""
def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE = False   )	->str:

  if not isinstance(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   ):
    a__:     Optional[int]										=	F'Expected string as input, found {type(_SCREAMING_SNAKE_CASE   )}'
    raise ValueError(_SCREAMING_SNAKE_CASE   )
  if not isinstance(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   ):
    a__:     List[str]										=	F'Expected boolean as use_pascal parameter, found {type(_SCREAMING_SNAKE_CASE   )}'
    raise ValueError(_SCREAMING_SNAKE_CASE   )

  a__:     int										=	input_str.split('_'   )

  a__:     List[str]										=	0 if use_pascal else 1

  a__:     List[str]										=	words[start_index:]

  a__:     List[str]										=	[word[0].upper() + word[1:] for word in words_to_capitalize]

  a__:     List[str]										=	'' if use_pascal else words[0]

  return "".join([initial_word, *capitalized_words]   )


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

 testmod()


