



from __future__ import annotations

from scipy.special import comb  # type: ignore



class   UpperCAmelCase_    :







							def __init__(   self,			__a):

														'''simple docstring'''

														_lowerCAmelCase     :							Optional[int]    =    list_of_points
														# Degree determines the flexibility of the curve.
														# Degree = 1 will produce a straight line.
														_lowerCAmelCase     :							Optional[Any]    =    len(__a) - 1







							def    snake_case__      (   self,			__a):

														'''simple docstring'''

														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,			__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    snake_case__      (   self,			__a):

														'''simple docstring'''

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

														_lowerCAmelCase     :							Tuple    =    self.basis_function(__a)
														_lowerCAmelCase     :							Any    =    0.0
														_lowerCAmelCase     :							Optional[int]    =    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,			__a = 0.01):

														'''simple docstring'''

														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     :							List[str]    =    0.0
														while t <= 1:
																					_lowerCAmelCase     :							int    =    self.bezier_curve_function(__a)
																					to_plot_x.append(value[0])
																					to_plot_y.append(value[1])
																					t += step_size

														_lowerCAmelCase     :							List[Any]    =    [i[0] for i in self.list_of_points]
														_lowerCAmelCase     :							Union[str, Any]    =    [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 colorsys

from PIL import Image  # type: ignore




def    __UpperCamelCase     (				lowerCAmelCase__							:				float  ,   lowerCAmelCase__							:				float  ,   lowerCAmelCase__							:				int  ):
 __a :  Any						=  x
 __a :  List[Any]						=  y
 for step in range(lowerCAmelCase__  ):  # noqa: B007
  __a :  List[Any]						=  a * a - b * b + x
  __a :  Tuple						=  2 * a * b + y
  __a :  Optional[int]						=  a_new

  # divergence happens for all complex number with an absolute value
  # greater than 4
  if a * a + b * b > 4:
   break
 return step / (max_step - 1)




def    __UpperCamelCase     (				lowerCAmelCase__							:				float  ):
 if distance == 1:
  return (0, 0, 0)
 else:
  return (2_5_5, 2_5_5, 2_5_5)




def    __UpperCamelCase     (				lowerCAmelCase__							:				float  ):
 if distance == 1:
  return (0, 0, 0)
 else:
  return tuple(round(i * 2_5_5  ) for i in colorsys.hsv_to_rgb(lowerCAmelCase__  ,   1  ,   1  )  )




def    __UpperCamelCase     (				lowerCAmelCase__							:				int = 8_0_0  ,   lowerCAmelCase__							:				int = 6_0_0  ,   lowerCAmelCase__							:				float = -0.6  ,   lowerCAmelCase__							:				float = 0  ,   lowerCAmelCase__							:				float = 3.2  ,   lowerCAmelCase__							:				int = 5_0  ,   lowerCAmelCase__							:				bool = True  ,   ):
 __a :  int						=  Image.new('''RGB'''  ,   (image_width, image_height)  )
 __a :  Dict						=  img.load()

 # loop through the image-coordinates
 for image_x in range(lowerCAmelCase__  ):
  for image_y in range(lowerCAmelCase__  ):
   # determine the figure-coordinates based on the image-coordinates
   __a :  Optional[Any]						=  figure_width / image_width * image_height
   __a :  str						=  figure_center_x + (image_x / image_width - 0.5) * figure_width
   __a :  str						=  figure_center_y + (image_y / image_height - 0.5) * figure_height

   __a :  Tuple						=  get_distance(lowerCAmelCase__  ,   lowerCAmelCase__  ,   lowerCAmelCase__  )

   # color the corresponding pixel based on the selected coloring-function
   if use_distance_color_coding:
    __a :  Optional[int]						=  get_color_coded_rgb(lowerCAmelCase__  )
   else:
    __a :  Optional[Any]						=  get_black_and_white_rgb(lowerCAmelCase__  )

 return img


if __name__ == "__main__":
      import doctest

      doctest.testmod()

      # colored version, full figure
      lowercase__       =get_image()

      # uncomment for colored version, different section, zoomed in
      # img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
      # figure_width = 0.8)

      # uncomment for black and white version, full figure
      # img = get_image(use_distance_color_coding = False)

      # uncomment to save the image
      # img.save("mandelbrot.png")

      img.show()
