




from __future__ import annotations

from scipy.special import comb  # type: ignore







class 						lowercase							:

     def __init__(  self		,     _a				)						->							Union[str, Any]:
           _A							:     Tuple			=						list_of_points
           # Degree determines the flexibility of the curve.
           # Degree = 1 will produce a straight line.
           _A							:     Dict			=						len(_a				) - 1

     def 					a__	(  self		,     _a				)						->							list[float]:
           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		,     _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 					a__	(  self		,     _a				)						->							tuple[float, float]:
           assert 0 <= t <= 1, "Time t must be between 0 and 1."

           _A							:     Optional[int]			=						self.basis_function(_a				)
           _A							:     List[str]			=						0.0
           _A							:     Tuple			=						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 					a__	(  self		,     _a = 0.01				)						->							str:
           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							:     str			=						0.0
           while t <= 1:
                 _A							:     Optional[Any]			=						self.bezier_curve_function(_a				)
                 to_plot_x.append(value[0]				)
                 to_plot_y.append(value[1]				)
                 t += step_size

           _A							:     Union[str, Any]			=						[i[0] for i in self.list_of_points]
           _A							:     str			=						[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









"""simple docstring"""




import unittest

import numpy as np

def 			lowerCAmelCase__     (     UpperCamelCase__     ,   UpperCamelCase__     ,   UpperCamelCase__     ,   UpperCamelCase__ = None     ,   ):



  '''simple docstring'''
  _a  :       List[Any]             =	np.shape(UpperCamelCase__				)
  _a  :       Any             =	np.shape(UpperCamelCase__				)
  _a  :       Union[str, Any]             =	np.shape(UpperCamelCase__				)

  if shape_a[0] != shape_b[0]:
    _a  :       int             =	(
        """Expected the same number of rows for A and B. """
        F"""Instead found A of size {shape_a} and B of size {shape_b}"""
    )
    raise ValueError(UpperCamelCase__				)

  if shape_b[1] != shape_c[1]:
    _a  :       Tuple             =	(
        """Expected the same number of columns for B and C. """
        F"""Instead found B of size {shape_b} and C of size {shape_c}"""
    )
    raise ValueError(UpperCamelCase__				)

  _a  :       int             =	pseudo_inv
  if a_inv is None:
    try:
      _a  :       Optional[int]             =	np.linalg.inv(UpperCamelCase__				)
    except np.linalg.LinAlgError:
      raise ValueError(
          """Input matrix A is not invertible. Cannot compute Schur complement."""				)

  return mat_c - mat_b.T @ a_inv @ mat_b






class        UpperCamelCase			(	unittest.TestCase      ):


 def _lowercase	(     self		:						int							)					->       None:
   _a  :       str             =	np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
   _a  :       Tuple             =	np.array([[0, 3], [3, 0], [2, 3]]							)
   _a  :       Optional[int]             =	np.array([[2, 1], [6, 3]]							)

   _a  :       Optional[Any]             =	schur_complement(UpperCAmelCase__							,						UpperCAmelCase__							,						UpperCAmelCase__							)

   _a  :       Union[str, Any]             =	np.block([[a, b], [b.T, c]]							)

   _a  :       int             =	np.linalg.det(UpperCAmelCase__							)
   _a  :       Union[str, Any]             =	np.linalg.det(UpperCAmelCase__							)
   _a  :       List[Any]             =	np.linalg.det(UpperCAmelCase__							)

   self.assertAlmostEqual(UpperCAmelCase__							,						det_a * det_s							)


 def _lowercase	(     self		:						int							)					->       None:
   _a  :       Optional[int]             =	np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
   _a  :       Optional[int]             =	np.array([[0, 3], [3, 0], [2, 3]]							)
   _a  :       Union[str, Any]             =	np.array([[2, 1], [6, 3]]							)

   with self.assertRaises(UpperCAmelCase__							):
     schur_complement(UpperCAmelCase__							,						UpperCAmelCase__							,						UpperCAmelCase__							)







 def _lowercase	(     self		:						List[Any]							)					->       None:
   _a  :       Any             =	np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
   _a  :       Dict             =	np.array([[0, 3], [3, 0], [2, 3]]							)
   _a  :       List[Any]             =	np.array([[2, 1, 3], [6, 3, 5]]							)

   with self.assertRaises(UpperCAmelCase__							):
     schur_complement(UpperCAmelCase__							,						UpperCAmelCase__							,						UpperCAmelCase__							)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 unittest.main()

