Computer Graphics Rendering import numpy as np from scipy.interpolate import CubicSpline import math
def smooth_attitude_interpolation(Cs, Cf, ωs, ωf, T): """ Smoothly interpolates between two attitude matrices Cs and Cf with rotations in any axis.
Args:
Cs (numpy.ndarray): The initial attitude matrix (3x3).
Cf (numpy.ndarray): The final attitude matrix (3x3).
ωs (numpy.ndarray): The initial angular velocity vector (3x1).
ωf (numpy.ndarray): The final angular velocity vector (3x1).
T (float): The time interval between Cs and Cf.
Returns:
List[numpy.ndarray]: A list of attitude matrices interpolating between Cs and Cf.
"""
if not np.allclose(np.linalg.inv(Cs) @ Cs, np.eye(3)):
raise ValueError("Cs is not a valid attitude matrix.")
if not np.allclose(np.linalg.inv(Cf) @ Cf, np.eye(3)):
raise ValueError("Cf is not a valid attitude matrix.")
t = np.linspace(0, T, 3)
def rotation_vector(t):
return np.log(Cs.T @ Cf)
θ_poly = CubicSpline(t, rotation_vector(t), bc_type=((1, 0.0), (1, 0.0)))
ω = θ_poly.derivative(nu=1)
ω_̇ = θ_poly.derivative(nu=2)
# Set jerk at endpoints to be equal
ω_̇(t[0]) = ω_̇(t[-1])
θ = np.array([ω(t_val) for t_val in t])
# Fit a cubic spline to the time matrix
t_poly = CubicSpline(t, np.exp(t), bc_type='not-a-knot')
# Interpolate attitude matrices
C = [Cs]
for i in range(len(t)):
C.append(Cs @ RY(2 * θ[i][0], axis=0) @ RX(2 * θ[i][1], axis=1) @ RZ(2 * θ[i][2], axis=2))
return C
def RY(θ, axis=0): """ Single-qubit Y-rotation gate around a specified axis.
Args:
θ (float): Rotation angle in radians.
axis (int): Axis of rotation (0 for X, 1 for Y, 2 for Z).
Returns:
numpy.ndarray: Single-qubit Y-rotation gate.
"""
if axis == 0:
return np.array([[math.cos(θ / 2), -math.sin(θ / 2)],
[math.sin(θ / 2), math.cos(θ / 2)]])
elif axis == 1:
return np.array([[math.cos(θ / 2), -math.sin(θ / 2)],
[math.sin(θ / 2), math.cos(θ / 2)]])
elif axis == 2:
return np.array([[math.cos(θ / 2), -1j * math.sin(θ / 2)],
[1j * math.sin(θ / 2), math.cos(θ / 2)]])
else:
raise ValueError("Invalid axis for RY gate.")
def RX(θ, axis=1): """ Single-qubit X-rotation gate around a specified axis.
Args:
θ (float): Rotation angle in radians.
axis (int): Axis of rotation (0 for X, 1 for Y, 2 for Z).
Returns:
numpy.ndarray: Single-qubit X-rotation gate.
"""
if axis == 0:
return np.array([[math.cos(θ / 2), -1j * math.sin(θ / 2)],
[-1j * math.sin(θ / 2), math.cos(θ / 2)]])
elif axis == 1:
return np.array([[math.cos(θ / 2), -math.sin(θ / 2)],
[math.sin(θ / 2), math.cos(θ / 2)]])
elif axis == 2:
return np.array([[math.cos(θ / 2), -math.sin(θ / 2)],
[math.sin(θ / 2), math.cos(θ / 2)]])
else:
raise ValueError("Invalid axis for RX gate.")
def RZ(θ, axis=2): """ Single-qubit Z-rotation gate around a specified axis.
Args:
θ (float): Rotation angle in radians.
axis (int): Axis of rotation (0 for X, 1 for Y, 2 for Z).
Returns:
numpy.ndarray: Single-qubit Z-rotation gate.
"""
if axis == 0:
return np.diag([math.exp(-1j * θ / 2), math.exp(1j * θ / 2)])
elif axis == 1:
return np.array([[math.cos(θ / 2), -1j * math.sin(θ / 2)],
[1j * math.sin(θ / 2), math.cos(θ / 2)]])
elif axis == 2:
return np.array([[math.cos(θ / 2), -math.sin(θ / 2)],
[math.sin(θ / 2), math.cos(θ / 2)]])
else:
raise ValueError("Invalid axis for RZ gate.")
def apply_quantum_gates(C, qubits): """ Applies the quantum gates in C to the qubits.
Args:
C (List[numpy.ndarray]): A list of quantum gates.
qubits (object): A quantum computing object (e.g., Qiskit Qubits).
"""
for gate in C:
qubits.unitary(gate, qubits.qubits)
return qubits