


'''simple docstring'''

from math import isclose, sqrt






def __UpperCamelCase   (			__lowerCamelCase		: float      ,  __lowerCamelCase		: float      ,  __lowerCamelCase		: float				)      ->     tuple[float, float, float]:







					'''simple docstring'''






					_a						=		point_y / 4 / point_x
					_a						=		2 * normal_gradient / (1 + normal_gradient * normal_gradient)
					_a						=		(1 - normal_gradient * normal_gradient) / (
					    1 + normal_gradient * normal_gradient
					)
					_a						=		(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
					_a						=		outgoing_gradient**2 + 4
					_a						=		2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
					_a						=		(point_y - outgoing_gradient * point_x) ** 2 - 100

					_a						=		(
					    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term				)
					) / (2 * quadratic_term)
					_a						=		(
					    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term				)
					) / (2 * quadratic_term)

					# two solutions, one of which is our input point
					_a						=		x_minus if isclose(__lowerCamelCase      ,  __lowerCamelCase				) else x_plus
					_a						=		point_y + outgoing_gradient * (next_x - point_x)

					return next_x, next_y, outgoing_gradient






def __UpperCamelCase   (			__lowerCamelCase		: float = 1.4      ,  __lowerCamelCase		: float = -9.6				)      ->     int:







					'''simple docstring'''






					_a						=		0
					_a						=		first_x_coord
					_a						=		first_y_coord
					_a						=		(10.1 - point_y) / (0.0 - point_x)

					while not (-0.01 <= point_x <= 0.01 and point_y > 0):
										_a			,			_a			,			_a						=		next_point(__lowerCamelCase      ,  __lowerCamelCase      ,  __lowerCamelCase				)
										num_reflections += 1

					return num_reflections


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





'''simple docstring'''

from collections import defaultdict

from graphs.minimum_spanning_tree_prims import prisms_algorithm as mst






def __UpperCamelCase   (			)      ->     Tuple:







     '''simple docstring'''






     _a			,			_a						=		9, 14  # noqa: F841
     _a						=		[
         [0, 1, 4],
         [0, 7, 8],
         [1, 2, 8],
         [7, 8, 7],
         [7, 6, 1],
         [2, 8, 2],
         [8, 6, 6],
         [2, 3, 7],
         [2, 5, 4],
         [6, 5, 2],
         [3, 5, 14],
         [3, 4, 9],
         [5, 4, 10],
         [1, 7, 11],
     ]

     _a						=		defaultdict(__lowerCamelCase				)
     for nodea, nodea, cost in edges:
          adjancency[nodea].append([nodea, cost]				)
          adjancency[nodea].append([nodea, cost]				)

     _a						=		mst(__lowerCamelCase				)

     _a						=		[
         [7, 6, 1],
         [2, 8, 2],
         [6, 5, 2],
         [0, 1, 4],
         [2, 5, 4],
         [2, 3, 7],
         [0, 7, 8],
         [3, 4, 9],
     ]

     for answer in expected:
          _a						=		tuple(answer[:2]				)
          _a						=		tuple(edge[::-1]				)
          assert edge in result or reverse in result

