Euler Angles¶

Three successive rotations about specified axes. The EulerAngle class stores three angles (phi, theta, psi) and a rotation order specifying which axes are used and in what sequence.

Rotation Orders¶

The 12 standard rotation sequences are defined by the EulerAngleOrder enum. Values are contiguous integers 0--11, suitable for use as branch indices in jax.lax.switch for JIT-compatible dispatch.

Symmetric Sequences¶

In symmetric (proper Euler) sequences, the first and third axes are the same:

Order	Axes	Index
`XYX`	X-Y-X	0
`XZX`	X-Z-X	2
`YXY`	Y-X-Y	4
`YZY`	Y-Z-Y	7
`ZXZ`	Z-X-Z	9
`ZYZ`	Z-Y-Z	11

Tait-Bryan Sequences¶

In Tait-Bryan sequences, all three axes are distinct:

Order	Axes	Index	Also known as
`XYZ`	X-Y-Z	1	Roll-Pitch-Yaw
`XZY`	X-Z-Y	3
`YXZ`	Y-X-Z	5
`YZX`	Y-Z-X	6
`ZXY`	Z-X-Y	8
`ZYX`	Z-Y-X	10	Yaw-Pitch-Roll

Construction¶

from astrojax import EulerAngle, EulerAngleOrder

# XYZ Tait-Bryan angles (roll-pitch-yaw)
e = EulerAngle(EulerAngleOrder.XYZ, 30.0, 45.0, 60.0, use_degrees=True)

# Angles in radians (default)
import math
e = EulerAngle(EulerAngleOrder.ZYX, math.pi/6, math.pi/4, math.pi/3)

From a Vector¶

import jax.numpy as jnp

vec = jnp.array([0.5236, 0.7854, 1.0472])  # radians
e = EulerAngle.from_vector(vec, EulerAngleOrder.XYZ)

Properties¶

Angles are stored in radians internally:

e = EulerAngle(EulerAngleOrder.XYZ, 30.0, 45.0, 60.0, use_degrees=True)
print(e.order)  # EulerAngleOrder.XYZ
print(e.phi)    # ~0.5236 rad
print(e.theta)  # ~0.7854 rad
print(e.psi)    # ~1.0472 rad

Order Conversion¶

Convert between different rotation orders by going through the quaternion representation:

e_xyz = EulerAngle(EulerAngleOrder.XYZ, 30.0, 45.0, 60.0, use_degrees=True)
e_zyx = e_xyz.to_euler_angle(EulerAngleOrder.ZYX)

Conversions¶

from astrojax import Quaternion, RotationMatrix, EulerAxis

e = EulerAngle(EulerAngleOrder.XYZ, 30.0, 45.0, 60.0, use_degrees=True)

# To other representations
q = e.to_quaternion()
r = e.to_rotation_matrix()
ea = e.to_euler_axis()

# From other representations
e2 = EulerAngle.from_quaternion(q, EulerAngleOrder.XYZ)
e3 = EulerAngle.from_rotation_matrix(r, EulerAngleOrder.XYZ)
e4 = EulerAngle.from_euler_axis(ea, EulerAngleOrder.XYZ)

Gimbal lock

Euler angles suffer from gimbal lock when the second rotation angle theta approaches \(\pm\pi/2\) (Tait-Bryan) or \(0\) / \(\pi\) (symmetric). Near these singularities, phi and psi become ambiguous. Consider using quaternions for singularity-free representation.