Elementary Rotations¶

The rotation_matrices module provides three functions -- Rx, Ry, Rz -- that construct 3x3 rotation matrices for rotations about the principal axes. These are the building blocks for composed rotations and are used internally by RotationMatrix.rotation_x(), rotation_y(), and rotation_z().

Functions¶

Rx -- Rotation About the X-Axis¶

\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta \\ 0 & -\sin\theta & \cos\theta \end{bmatrix} \]

from astrojax import Rx

# Angle in radians (default)
r = Rx(0.7854)

# Angle in degrees
r = Rx(45.0, use_degrees=True)

Ry -- Rotation About the Y-Axis¶

\[ R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix} \]

from astrojax import Ry

r = Ry(30.0, use_degrees=True)

Rz -- Rotation About the Z-Axis¶

\[ R_z(\theta) = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

from astrojax import Rz

r = Rz(60.0, use_degrees=True)

Composing Rotations¶

Elementary rotations can be composed by matrix multiplication to build arbitrary rotation sequences:

import jax.numpy as jnp
from astrojax import Rx, Ry, Rz

# ZYX Euler rotation sequence
R = Rz(60.0, use_degrees=True) @ Ry(30.0, use_degrees=True) @ Rx(45.0, use_degrees=True)

# Rotate a vector
v = jnp.array([1.0, 0.0, 0.0])
v_rotated = R @ v

Return type

Rx, Ry, and Rz return raw jnp.ndarray arrays (shape (3, 3)), not RotationMatrix instances. Use RotationMatrix.rotation_x() etc. if you need the class wrapper.

Convention¶

The rotation direction follows the right-hand rule: a positive angle produces a counter-clockwise rotation when viewed from the positive end of the axis looking toward the origin.