

from __future__ import annotations

from scipy.special import comb  # type: ignore







class        a__       :







  """simple docstring"""




  def __init__(			self    : List[Any]				,				UpperCAmelCase__    : List[str]		)					->List[Any]:
        """simple docstring"""







        SCREAMING_SNAKE_CASE :      Optional[Any]       =   list_of_points
        # Degree determines the flexibility of the curve.
        # Degree = 1 will produce a straight line.
        SCREAMING_SNAKE_CASE :      Union[str, Any]       =   len(lowerCAmelCase_		) - 1


  def  _lowercase      (			self    : Any				,				UpperCAmelCase__    : Union[str, Any]		)					->list[float]:
        """simple docstring"""







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


  def  _lowercase      (			self    : List[str]				,				UpperCAmelCase__    : Dict		)					->tuple[float, float]:
        """simple docstring"""







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

        SCREAMING_SNAKE_CASE :      Optional[int]       =   self.basis_function(lowerCAmelCase_		)
        SCREAMING_SNAKE_CASE :      List[str]       =   0.0
        SCREAMING_SNAKE_CASE :      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  _lowercase      (			self    : Dict				,				UpperCAmelCase__    : Optional[int] = 0.01		)					->Optional[int]:
        """simple docstring"""







        from matplotlib import pyplot as plt  # type: ignore

        SCREAMING_SNAKE_CASE :      int       =   []  # x coordinates of points to plot
        SCREAMING_SNAKE_CASE :      Tuple       =   []  # y coordinates of points to plot

        SCREAMING_SNAKE_CASE :      Optional[int]       =   0.0
        while t <= 1:
              SCREAMING_SNAKE_CASE :      Union[str, Any]       =   self.bezier_curve_function(lowerCAmelCase_		)
              to_plot_x.append(value[0]		)
              to_plot_y.append(value[1]		)
              t += step_size

        SCREAMING_SNAKE_CASE :      str       =   [i[0] for i in self.list_of_points]
        SCREAMING_SNAKE_CASE :      str       =   [i[1] for i in self.list_of_points]

        plt.plot(
            lowerCAmelCase_				,				lowerCAmelCase_				,				color="""blue"""				,				label="""Curve of Degree """ + str(self.degree		)				,				)
        plt.scatter(lowerCAmelCase_				,				lowerCAmelCase_				,				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 					__lowercase      (     _A			)     ->						list[int]:
      SCREAMING_SNAKE_CASE :      int       =   [0 for i in range(len(_A			)			)]

      # initialize interval's left pointer and right pointer
      SCREAMING_SNAKE_CASE     ,	SCREAMING_SNAKE_CASE :      Dict       =   0, 0

      for i in range(1						,    len(_A			)			):
            # case when current index is inside the interval
            if i <= right_pointer:
                  SCREAMING_SNAKE_CASE :      Any       =   min(right_pointer - i + 1						,    z_result[i - left_pointer]			)
                  SCREAMING_SNAKE_CASE :      Optional[Any]       =   min_edge

            while go_next(_A						,    _A						,    _A			):
                  z_result[i] += 1

            # if new index's result gives us more right interval,
            # we've to update left_pointer and right_pointer
            if i + z_result[i] - 1 > right_pointer:
                  SCREAMING_SNAKE_CASE     ,	SCREAMING_SNAKE_CASE :      List[Any]       =   i, i + z_result[i] - 1

      return z_result






def 					__lowercase      (     _A						,    _A						,    _A			)     ->						bool:
      return i + z_result[i] < len(_A			) and s[z_result[i]] == s[i + z_result[i]]






def 					__lowercase      (     _A						,    _A			)     ->						int:
      SCREAMING_SNAKE_CASE :      Any       =   0
      # concatenate 'pattern' and 'input_str' and call z_function
      # with concatenated string
      SCREAMING_SNAKE_CASE :      List[Any]       =   z_function(pattern + input_str			)

      for val in z_result:
            # if value is greater then length of the pattern string
            # that means this index is starting position of substring
            # which is equal to pattern string
            if val >= len(_A			):
                  answer += 1

      return answer


if __name__ == "__main__":
  import doctest

  doctest.testmod()




