Coordinate Transforms Module#

The coordinates module provides functions for converting between various coordinate systems used in antenna and radar applications, including antenna coordinates, radar coordinates, cone/clock coordinates, and rotation matrices for pattern rotation.

Antenna and Radar Coordinates#

Functions for converting between antenna (theta/phi) and radar (az/el) coordinate systems.

phased_array.antenna_to_radar(theta_ant, phi_ant)[source]#

Convert antenna coordinates (theta/phi) to radar coordinates (az/el).

Antenna coordinates:

  • theta: angle from boresight (z-axis), 0 at boresight

  • phi: azimuthal angle in x-y plane, 0 along x-axis

Radar coordinates:

  • az: azimuth in horizontal plane, 0 at boresight

  • el: elevation from horizon, 0 at horizon, positive up

Parameters:

  • theta_ant (array_like) – Theta angle in radians (from boresight)

  • phi_ant (array_like) – Phi angle in radians (azimuthal)

Returns:

  • az (ndarray) – Azimuth angle in radians

  • el (ndarray) – Elevation angle in radians

Return type:

Tuple[ndarray, ndarray]

Examples

Boresight in antenna coords is also boresight in radar:

>>> import numpy as np
>>> import phased_array as pa
>>> az, el = pa.antenna_to_radar(0.0, 0.0)
>>> np.isclose(az, 0.0) and np.isclose(el, np.pi/2)
True

Off-boresight conversion:

>>> az, el = pa.antenna_to_radar(np.pi/6, 0.0)  # 30 deg in x-z plane
>>> np.isclose(np.rad2deg(az), 0.0, atol=1e-10)
True
phased_array.radar_to_antenna(az, el)[source]#

Convert radar coordinates (az/el) to antenna coordinates (theta/phi).

Parameters:

  • az (array_like) – Azimuth angle in radians

  • el (array_like) – Elevation angle in radians (from horizon)

Returns:

  • theta (ndarray) – Theta angle in radians (from boresight)

  • phi (ndarray) – Phi angle in radians (azimuthal)

Return type:

Tuple[ndarray, ndarray]

Examples

>>> import numpy as np
>>> import phased_array as pa
>>> theta, phi = pa.radar_to_antenna(0.0, np.pi/2)  # Boresight
>>> np.isclose(theta, 0.0, atol=1e-10)
True

Round-trip conversion:

>>> theta_orig, phi_orig = np.pi/4, np.pi/3
>>> az, el = pa.antenna_to_radar(theta_orig, phi_orig)
>>> theta, phi = pa.radar_to_antenna(az, el)
>>> np.isclose(theta, theta_orig, atol=1e-10)
True

Cone/Clock Coordinates#

Cone/clock coordinates are useful for describing patterns on aircraft radomes or for visualizing scan limits.

phased_array.antenna_to_cone(theta, phi)[source]#

Convert antenna coordinates (theta/phi) to cone/clock coordinates.

Cone/clock coordinates are useful for describing patterns on aircraft radomes or for visualizing scan limits.

Parameters:

  • theta (array_like) – Theta angle in radians (from boresight)

  • phi (array_like) – Phi angle in radians (azimuthal)

Returns:

  • cone (ndarray) – Cone angle in radians (distance from boresight, same as theta)

  • clock (ndarray) – Clock angle in radians (azimuthal position, same as phi)

Return type:

Tuple[ndarray, ndarray]

Examples

>>> import numpy as np
>>> import phased_array as pa
>>> cone, clock = pa.antenna_to_cone(np.pi/6, np.pi/4)
>>> np.isclose(cone, np.pi/6) and np.isclose(clock, np.pi/4)
True
phased_array.cone_to_antenna(cone, clock)[source]#

Convert cone/clock coordinates to antenna coordinates (theta/phi).

Parameters:

  • cone (array_like) – Cone angle in radians

  • clock (array_like) – Clock angle in radians

Returns:

  • theta (ndarray) – Theta angle in radians

  • phi (ndarray) – Phi angle in radians

Return type:

Tuple[ndarray, ndarray]

Examples

>>> import numpy as np
>>> import phased_array as pa
>>> theta, phi = pa.cone_to_antenna(np.pi/6, np.pi/4)
>>> np.isclose(theta, np.pi/6) and np.isclose(phi, np.pi/4)
True

Rotation Matrices#

3x3 rotation matrices for transforming coordinate systems using aircraft-style roll, pitch, and yaw angles.

phased_array.rotation_matrix_roll(angle)[source]#

Create 3x3 rotation matrix for roll (rotation about x-axis).

Parameters:

angle (float) – Roll angle in radians (positive = right wing down)

Returns:

R – 3x3 rotation matrix

Return type:

ndarray

Examples

>>> import numpy as np
>>> import phased_array as pa
>>> R = pa.rotation_matrix_roll(0.0)
>>> np.allclose(R, np.eye(3))
True

90 degree roll:

>>> R = pa.rotation_matrix_roll(np.pi/2)
>>> v = np.array([0, 1, 0])  # y-axis
>>> v_rot = R @ v
>>> np.allclose(v_rot, [0, 0, 1], atol=1e-10)  # Rotates to z-axis
True
phased_array.rotation_matrix_pitch(angle)[source]#

Create 3x3 rotation matrix for pitch (rotation about y-axis).

Parameters:

angle (float) – Pitch angle in radians (positive = nose up)

Returns:

R – 3x3 rotation matrix

Return type:

ndarray

Examples

>>> import numpy as np
>>> import phased_array as pa
>>> R = pa.rotation_matrix_pitch(0.0)
>>> np.allclose(R, np.eye(3))
True

90 degree pitch:

>>> R = pa.rotation_matrix_pitch(np.pi/2)
>>> v = np.array([0, 0, 1])  # z-axis (boresight)
>>> v_rot = R @ v
>>> np.allclose(v_rot, [1, 0, 0], atol=1e-10)  # Rotates to x-axis
True
phased_array.rotation_matrix_yaw(angle)[source]#

Create 3x3 rotation matrix for yaw (rotation about z-axis).

Parameters:

angle (float) – Yaw angle in radians (positive = nose left)

Returns:

R – 3x3 rotation matrix

Return type:

ndarray

Examples

>>> import numpy as np
>>> import phased_array as pa
>>> R = pa.rotation_matrix_yaw(0.0)
>>> np.allclose(R, np.eye(3))
True

90 degree yaw:

>>> R = pa.rotation_matrix_yaw(np.pi/2)
>>> v = np.array([1, 0, 0])  # x-axis
>>> v_rot = R @ v
>>> np.allclose(v_rot, [0, 1, 0], atol=1e-10)  # Rotates to y-axis
True

Pattern Rotation#

Functions for rotating radiation patterns in 3D space.

phased_array.rotate_pattern(theta, phi, pattern, roll_deg, pitch_deg, yaw_deg)[source]#

Rotate a radiation pattern by specified Euler angles.

The rotation order is: yaw -> pitch -> roll (intrinsic rotations), which corresponds to standard aerospace convention.

Parameters:

  • theta (ndarray) – Original theta angles in radians (1D or 2D grid)

  • phi (ndarray) – Original phi angles in radians (same shape as theta)

  • pattern (ndarray) – Pattern values (complex or magnitude, same shape as theta)

  • roll_deg (float) – Roll angle in degrees

  • pitch_deg (float) – Pitch angle in degrees

  • yaw_deg (float) – Yaw angle in degrees

Returns:

  • theta_new (ndarray) – New theta coordinates after rotation

  • phi_new (ndarray) – New phi coordinates after rotation

  • pattern_interp (ndarray) – Pattern values interpolated onto the new grid

Return type:

Tuple[ndarray, ndarray, ndarray]

Examples

Rotate pattern by 30 degrees in yaw:

>>> import numpy as np
>>> import phased_array as pa
>>> theta = np.linspace(0, np.pi/2, 46)
>>> phi = np.linspace(0, 2*np.pi, 73)
>>> theta_grid, phi_grid = np.meshgrid(theta, phi, indexing='ij')
>>> pattern = np.cos(theta_grid)  # Simple cosine pattern
>>> theta_r, phi_r, pattern_r = pa.rotate_pattern(
...     theta_grid, phi_grid, pattern,
...     roll_deg=0, pitch_deg=0, yaw_deg=30
... )
>>> pattern_r.shape == pattern.shape
True

Notes

For points that rotate outside the original grid, extrapolation may produce artifacts. Consider padding the original pattern.