

from __future__ import annotations

from scipy.special import comb  # type: ignore


class  A						:







    """simple docstring"""





    def __init__(      self :       Tuple,lowercase_ :       list[tuple[float, float]]		)->    Tuple:





        '''simple docstring'''







        A__          =   list_of_points
        # Degree determines the flexibility of the curve.
        # Degree = 1 will produce a straight line.
        A__          =   len(lowercase_		) - 1


    def 						snake_case__     (      self :       Dict,lowercase_ :       float		)->    list[float]:





        '''simple docstring'''







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


    def 						snake_case__     (      self :       Union[str, Any],lowercase_ :       float		)->    tuple[float, float]:





        '''simple docstring'''







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

        A__          =   self.basis_function(lowercase_		)
        A__          =   0.0
        A__          =   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 :       Dict,lowercase_ :       float = 0.01		)->    Optional[Any]:





        '''simple docstring'''







        from matplotlib import pyplot as plt  # type: ignore

        A__          =   []  # x coordinates of points to plot
        A__          =   []  # y coordinates of points to plot

        A__          =   0.0
        while t <= 1:
            A__          =   self.bezier_curve_function(lowercase_		)
            to_plot_x.append(value[0]		)
            to_plot_y.append(value[1]		)
            t += step_size

        A__          =   [i[0] for i in self.list_of_points]
        A__          =   [i[1] for i in self.list_of_points]

        plt.plot(
            lowercase_,lowercase_,color='blue',label='Curve of Degree ' + str(self.degree		),)
        plt.scatter(lowercase_,lowercase_,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


from __future__ import annotations





class      A							:


   def __init__(self     :						Union[str, Any]       ,  __UpperCAmelCase     :						list[list[int]]		)						->				List[str]:
         """simple docstring"""



         UpperCAmelCase__								=  TypeError(
             "Matrices must be formed from a list of zero or more lists containing at "
             "least one and the same number of values, each of which must be of type "
             "int or float."		)
         if len(__UpperCAmelCase		) != 0:
               UpperCAmelCase__								=  len(rows[0]		)
               if cols == 0:
                     raise error
               for row in rows:
                     if len(__UpperCAmelCase		) != cols:
                           raise error
                     for value in row:
                           if not isinstance(__UpperCAmelCase       ,  (int, float)		):
                                 raise error
               UpperCAmelCase__								=  rows
         else:
               UpperCAmelCase__								=  []


   def 			lowercase_      (self     :						Any		)						->				list[list[int]]:
         """simple docstring"""



         return [[row[i] for row in self.rows] for i in range(len(self.rows[0]		)		)]


   @property
   def 			lowercase_      (self     :						Any		)						->				int:
         """simple docstring"""



         return len(self.rows		)


   @property
   def 			lowercase_      (self     :						Union[str, Any]		)						->				int:
         """simple docstring"""



         return len(self.rows[0]		)


   @property
   def 			lowercase_      (self     :						List[Any]		)						->				tuple[int, int]:
         """simple docstring"""



         return (self.num_rows, self.num_columns)


   @property
   def 			lowercase_      (self     :						Tuple		)						->				bool:
         """simple docstring"""



         return self.order[0] == self.order[1]


   def 			lowercase_      (self     :						Any		)						->				Matrix:
         """simple docstring"""



         UpperCAmelCase__								=  [
             [0 if column_num != row_num else 1 for column_num in range(self.num_rows		)]
             for row_num in range(self.num_rows		)
         ]
         return Matrix(__UpperCAmelCase		)


   def 			lowercase_      (self     :						int		)						->				int:
         """simple docstring"""



         if not self.is_square:
               return 0
         if self.order == (0, 0):
               return 1
         if self.order == (1, 1):
               return int(self.rows[0][0]		)
         if self.order == (2, 2):
               return int(
                   (self.rows[0][0] * self.rows[1][1])
                   - (self.rows[0][1] * self.rows[1][0])		)
         else:
               return sum(
                   self.rows[0][column] * self.cofactors().rows[0][column]
                   for column in range(self.num_columns		)		)


   def 			lowercase_      (self     :						Tuple		)						->				bool:
         """simple docstring"""



         return bool(self.determinant()		)


   def 			lowercase_      (self     :						Dict       ,  __UpperCAmelCase     :						int       ,  __UpperCAmelCase     :						int		)						->				int:
         """simple docstring"""



         UpperCAmelCase__								=  [
             [
                 self.rows[other_row][other_column]
                 for other_column in range(self.num_columns		)
                 if other_column != column
             ]
             for other_row in range(self.num_rows		)
             if other_row != row
         ]
         return Matrix(__UpperCAmelCase		).determinant()


   def 			lowercase_      (self     :						int       ,  __UpperCAmelCase     :						int       ,  __UpperCAmelCase     :						int		)						->				int:
         """simple docstring"""



         if (row + column) % 2 == 0:
               return self.get_minor(__UpperCAmelCase       ,  __UpperCAmelCase		)
         return -1 * self.get_minor(__UpperCAmelCase       ,  __UpperCAmelCase		)


   def 			lowercase_      (self     :						Union[str, Any]		)						->				Matrix:
         """simple docstring"""



         return Matrix(
             [
                 [self.get_minor(__UpperCAmelCase       ,  __UpperCAmelCase		) for column in range(self.num_columns		)]
                 for row in range(self.num_rows		)
             ]		)


   def 			lowercase_      (self     :						List[str]		)						->				Matrix:
         """simple docstring"""



         return Matrix(
             [
                 [
                     self.minors().rows[row][column]
                     if (row + column) % 2 == 0
                     else self.minors().rows[row][column] * -1
                     for column in range(self.minors().num_columns		)
                 ]
                 for row in range(self.minors().num_rows		)
             ]		)


   def 			lowercase_      (self     :						Optional[Any]		)						->				Matrix:
         """simple docstring"""



         UpperCAmelCase__								=  [
             [self.cofactors().rows[column][row] for column in range(self.num_columns		)]
             for row in range(self.num_rows		)
         ]
         return Matrix(__UpperCAmelCase		)


   def 			lowercase_      (self     :						List[Any]		)						->				Matrix:
         """simple docstring"""



         UpperCAmelCase__								=  self.determinant()
         if not determinant:
               raise TypeError("Only matrices with a non-zero determinant have an inverse"		)
         return self.adjugate() * (1 / determinant)


   def __repr__(self     :						Dict		)						->				str:
         """simple docstring"""



         return str(self.rows		)


   def __str__(self     :						Optional[Any]		)						->				str:
         """simple docstring"""



         if self.num_rows == 0:
               return "[]"
         if self.num_rows == 1:
               return "[[" + ". ".join(str(self.rows[0]		)		) + "]]"
         return (
             "["
             + "\n ".join(
                 [
                     "[" + ". ".join([str(__UpperCAmelCase		) for value in row]		) + ".]"
                     for row in self.rows
                 ]		)
             + "]"
         )


   def 			lowercase_      (self     :						Optional[int]       ,  __UpperCAmelCase     :						list[int]       ,  __UpperCAmelCase     :						int | None = None		)						->				None:
         """simple docstring"""



         UpperCAmelCase__								=  TypeError("Row must be a list containing all ints and/or floats"		)
         if not isinstance(__UpperCAmelCase       ,  __UpperCAmelCase		):
               raise type_error
         for value in row:
               if not isinstance(__UpperCAmelCase       ,  (int, float)		):
                     raise type_error
         if len(__UpperCAmelCase		) != self.num_columns:
               raise ValueError(
                   "Row must be equal in length to the other rows in the matrix"		)
         if position is None:
               self.rows.append(__UpperCAmelCase		)
         else:
               UpperCAmelCase__								=  self.rows[0:position] + [row] + self.rows[position:]


   def 			lowercase_      (self     :						Union[str, Any]       ,  __UpperCAmelCase     :						list[int]       ,  __UpperCAmelCase     :						int | None = None		)						->				None:
         """simple docstring"""



         UpperCAmelCase__								=  TypeError(
             "Column must be a list containing all ints and/or floats"		)
         if not isinstance(__UpperCAmelCase       ,  __UpperCAmelCase		):
               raise type_error
         for value in column:
               if not isinstance(__UpperCAmelCase       ,  (int, float)		):
                     raise type_error
         if len(__UpperCAmelCase		) != self.num_rows:
               raise ValueError(
                   "Column must be equal in length to the other columns in the matrix"		)
         if position is None:
               UpperCAmelCase__								=  [self.rows[i] + [column[i]] for i in range(self.num_rows		)]
         else:
               UpperCAmelCase__								=  [
                   self.rows[i][0:position] + [column[i]] + self.rows[i][position:]
                   for i in range(self.num_rows		)
               ]


   def __eq__(self     :						Any       ,  __UpperCAmelCase     :						object		)						->				bool:
         """simple docstring"""



         if not isinstance(__UpperCAmelCase       ,  __UpperCAmelCase		):
               return NotImplemented
         return self.rows == other.rows


   def __ne__(self     :						int       ,  __UpperCAmelCase     :						object		)						->				bool:
         """simple docstring"""



         return not self == other


   def __neg__(self     :						Dict		)						->				Matrix:
         """simple docstring"""



         return self * -1


   def __add__(self     :						Dict       ,  __UpperCAmelCase     :						Matrix		)						->				Matrix:
         """simple docstring"""



         if self.order != other.order:
               raise ValueError("Addition requires matrices of the same order"		)
         return Matrix(
             [
                 [self.rows[i][j] + other.rows[i][j] for j in range(self.num_columns		)]
                 for i in range(self.num_rows		)
             ]		)


   def __sub__(self     :						Optional[Any]       ,  __UpperCAmelCase     :						Matrix		)						->				Matrix:
         """simple docstring"""



         if self.order != other.order:
               raise ValueError("Subtraction requires matrices of the same order"		)
         return Matrix(
             [
                 [self.rows[i][j] - other.rows[i][j] for j in range(self.num_columns		)]
                 for i in range(self.num_rows		)
             ]		)


   def __mul__(self     :						Tuple       ,  __UpperCAmelCase     :						Matrix | int | float		)						->				Matrix:
         """simple docstring"""



         if isinstance(__UpperCAmelCase       ,  (int, float)		):
               return Matrix(
                   [[int(element * other		) for element in row] for row in self.rows]		)
         elif isinstance(__UpperCAmelCase       ,  __UpperCAmelCase		):
               if self.num_columns != other.num_rows:
                     raise ValueError(
                         "The number of columns in the first matrix must "
                         "be equal to the number of rows in the second"		)
               return Matrix(
                   [
                       [Matrix.dot_product(__UpperCAmelCase       ,  __UpperCAmelCase		) for column in other.columns()]
                       for row in self.rows
                   ]		)
         else:
               raise TypeError(
                   "A Matrix can only be multiplied by an int, float, or another matrix"		)


   def __pow__(self     :						List[Any]       ,  __UpperCAmelCase     :						int		)						->				Matrix:
         """simple docstring"""



         if not isinstance(__UpperCAmelCase       ,  __UpperCAmelCase		):
               raise TypeError("A Matrix can only be raised to the power of an int"		)
         if not self.is_square:
               raise ValueError("Only square matrices can be raised to a power"		)
         if other == 0:
               return self.identity()
         if other < 0:
               if self.is_invertable():
                     return self.inverse() ** (-other)
               raise ValueError(
                   "Only invertable matrices can be raised to a negative power"		)
         UpperCAmelCase__								=  self
         for _ in range(other - 1		):
               result *= self
         return result





   @classmethod
   def 			lowercase_      (cls     :						Dict       ,  __UpperCAmelCase     :						list[int]       ,  __UpperCAmelCase     :						list[int]		)						->				int:
         """simple docstring"""



         return sum(row[i] * column[i] for i in range(len(__UpperCAmelCase		)		)		)


if __name__ == "__main__":
      import doctest

      doctest.testmod()



