





from PIL import Image






def 					lowerCamelCase_						(    _UpperCamelCase			)		->						Image:






				"""simple docstring"""


				snake_case_      ,							snake_case_						:   List[Any]           =  image.size
				snake_case_						:   Optional[Any]           =  0
				snake_case_						:   str           =  image.load()
				for i in range(_UpperCamelCase			):
								for j in range(_UpperCamelCase			):
												snake_case_						:   Union[str, Any]           =  pixels[j, i]
												mean += pixel
				mean //= width * height

				for j in range(_UpperCamelCase			):
								for i in range(_UpperCamelCase			):
												snake_case_						:   List[str]           =  255 if pixels[i, j] > mean else 0
				return image


if __name__ == "__main__":
	lowerCAmelCase_    =    mean_threshold(Image.open('''path_to_image''').convert('''L'''))
	image.save('''output_image_path''')







from math import isclose, sqrt






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






				"""simple docstring"""


				snake_case_						:   Dict           =  point_y / 4 / point_x
				snake_case_						:   List[str]           =  2 * normal_gradient / (1 + normal_gradient * normal_gradient)
				snake_case_						:   Union[str, Any]           =  (1 - normal_gradient * normal_gradient) / (
				    1 + normal_gradient * normal_gradient
				)
				snake_case_						:   Tuple           =  (sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

				# to find the next point, solve the simultaeneous equations:
				# y^2 + 4x^2 = 100
				# y - b = m * (x - a)
				# ==> A x^2 + B x + C = 0
				snake_case_						:   Union[str, Any]           =  outgoing_gradient**2 + 4
				snake_case_						:   Tuple           =  2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
				snake_case_						:   Optional[Any]           =  (point_y - outgoing_gradient * point_x) ** 2 - 100

				snake_case_						:   Dict           =  (
				    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term			)
				) / (2 * quadratic_term)
				snake_case_						:   Optional[int]           =  (
				    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term			)
				) / (2 * quadratic_term)

				# two solutions, one of which is our input point
				snake_case_						:   Any           =  x_minus if isclose(_UpperCamelCase			,    _UpperCamelCase			) else x_plus
				snake_case_						:   int           =  point_y + outgoing_gradient * (next_x - point_x)

				return next_x, next_y, outgoing_gradient






def 					lowerCamelCase_						(    _UpperCamelCase = 1.4			,    _UpperCamelCase = -9.6			)		->						int:






				"""simple docstring"""


				snake_case_						:   int           =  0
				snake_case_						:   float           =  first_x_coord
				snake_case_						:   float           =  first_y_coord
				snake_case_						:   float           =  (10.1 - point_y) / (0.0 - point_x)

				while not (-0.01 <= point_x <= 0.01 and point_y > 0):
								snake_case_      ,							snake_case_      ,							snake_case_						:   str           =  next_point(_UpperCamelCase			,    _UpperCamelCase			,    _UpperCamelCase			)
								num_reflections += 1

				return num_reflections


if __name__ == "__main__":
	print(F'''{solution() = }''')
