Rotation Representations: A Cross-Reference Guide
Overview
A rotation describes an orientation or an orientation change. It does not, by itself, describe the path taken, the location of a joint, the forces that produced the motion, or the energy transferred. These distinctions matter when comparing pelvis and thorax motion, interpreting a clubface angle, or building a humanoid model.
This reference and the interactive converter use right-handed active rotations of column vectors. For a vector rotated within one coordinate system, \(v'=Rv\). For coordinates of the same physical vector in different frames, \(v^A=R_{AB}v^B\): the columns of \(R_{AB}\) are the axes of frame \(B\) expressed in \(A\). These interpretations share the same algebra; declare which one a measurement or calculation uses. All mathematical angles below are in radians. Quaternions use Hamilton multiplication and scalar-first order \((w,x,y,z)\).
| Representation | Numbers Stored | Constraints and Ambiguities |
|---|---|---|
| \(R\in SO(3)\) | 9 | \(R^TR=I\), \(\det R=1\); three independent degrees of freedom |
| Unit quaternion \(q\in S^3\) | 4 | Unit norm; \(q\) and \(-q\) give the same orientation |
| Euler angles | 3 | Axis order and moving/fixed-axis convention are essential; singular coordinate charts |
| Axis-angle \((n,\theta)\) | 4 | Unit axis; identity axis undefined; equivalent angle/axis choices |
| Rotation vector \(\rho=\theta n\in\mathbb R^3\) | 3 | \([\rho]_\times\in\mathfrak{so}(3)\); logarithm branches and a half-turn ambiguity |
The number of stored scalars is not the number of physical degrees of freedom. A unit quaternion avoids Euler gimbal lock, but choosing a unique quaternion from every orientation cannot also give a globally continuous choice.
Rotation Matrices
The special orthogonal group is
\[SO(3)=\{R\in\mathbb R^{3\times3}:R^TR=I,\ \det R=1\}.\]
Orthogonality alone also admits reflections with determinant \(-1\). A reflection is not a rotation. Composition follows frame cancellation:
\[v^A=R_{AB}(R_{BC}v^C),\qquad R_{AC}=R_{AB}R_{BC},\qquad R_{BA}=R_{AB}^T.\]
The rightmost transformation acts first. In a single frame, \(R_z(\pi/2)R_x(\pi/2)e_y=e_z\), whereas reversing the factors gives \(R_x(\pi/2)R_z(\pi/2)e_y=-e_x\). The two valid compositions describe different motions.
Standard Rotation Matrices
\[R_x(\alpha)=\begin{bmatrix}1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha\end{bmatrix},\]
\[R_y(\beta)=\begin{bmatrix}\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta\end{bmatrix},\]
\[R_z(\gamma)=\begin{bmatrix}\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1\end{bmatrix}.\]
For any \(a,b\), \([a]_\times b=a\times b\), where
\[[a]_\times=\begin{bmatrix}0&-a_z&a_y\\a_z&0&-a_x\\-a_y&a_x&0\end{bmatrix}.\]
Euler Angles
Intrinsic ZYX means yaw about the initial \(z\) axis, pitch about the resulting \(y\) axis, then roll about the resulting \(x\) axis:
\[R=R_z(\psi)R_y(\theta)R_x(\phi).\]
It is equivalent to fixed-axis XYZ rotations applied in the temporal order roll, pitch, yaw. Calling either sequence merely “ZYX” without saying which axes move is incomplete. Here \((\psi,\theta,\phi)\) always means yaw, pitch, roll in that matrix product.
Converting Euler Angles to Rotation Matrix
With \(c_a=\cos a\) and \(s_a=\sin a\),
\[R=\begin{bmatrix} c_\psi c_\theta&c_\psi s_\theta s_\phi-s_\psi c_\phi&c_\psi s_\theta c_\phi+s_\psi s_\phi\\ s_\psi c_\theta&s_\psi s_\theta s_\phi+c_\psi c_\phi&s_\psi s_\theta c_\phi-c_\psi s_\phi\\ -s_\theta&c_\theta s_\phi&c_\theta c_\phi \end{bmatrix}.\]
On the principal chart \(-\pi/2<\theta<\pi/2\),
\[\psi=\operatorname{atan2}(R_{21},R_{11}),\qquad \phi=\operatorname{atan2}(R_{32},R_{33}),\]
\[\theta=\operatorname{atan2}\!\left(-R_{31},\sqrt{R_{11}^2+R_{21}^2}\right).\]
The familiar \(-\arcsin(R_{31})\) is algebraically equivalent on this chart for an exact rotation. The atan2 form retains information in the small cosine terms when \(R_{31}\) rounds to \(\pm1\).
At ZYX pitch \(+\pi/2\), orientation determines \(\psi-\phi\); at \(-\pi/2\), it determines \(\psi+\phi\). Choose \(\phi=0\) and \(\psi=\operatorname{atan2}(-R_{12},R_{22})\) at either lock. The rigid body retains its physical rotational freedom; the three-angle chart loses local independence.
The condition depends on the convention. Proper Euler ZYZ, \(R_z(\alpha)R_y(\beta)R_z(\gamma)\), locks at \(\beta=0\) or \(\pi\), not at \(\pi/2\). Near a chart singularity, coordinate uncertainty can be large even when orientation uncertainty is small. Changing representation does not remove sensor noise.
Euler Rates and Angular Velocity
Differentiating the declared product gives spatial angular velocity
\[\omega_s=\begin{bmatrix} 0&-\sin\psi&\cos\psi\cos\theta\\ 0&\cos\psi&\sin\psi\cos\theta\\ 1&0&-\sin\theta \end{bmatrix} \begin{bmatrix}\dot\psi\\\dot\theta\\\dot\phi\end{bmatrix}.\]
The columns are the successive rotation axes expressed in the same frame. Their determinant magnitude is \(|\cos\theta|\). At yaw \(\pi/2\), pitch zero and only \(\dot\phi=1\) rad/s, \(\omega_s=(0,1,0)\) rad/s: a roll-coordinate rate is not automatically the world \(x\) component. Peak Euler-coordinate rates from different conventions therefore cannot be treated as interchangeable segment-speed measurements.
Axis-Angle Representation
For a unit axis \(n\) and angle \(\theta\),
\[R=I+\sin\theta[n]_\times+(1-\cos\theta)[n]_\times^2 =\exp(\theta[n]_\times).\]
This is Rodrigues’ formula. The identity \(1-\cos\theta=2\sin^2(\theta/2)\) is useful for small angles. Any nonzero input axis can first be normalized; a zero axis cannot define a nonzero rotation.
A principal representation uses \(0\leq\theta\leq\pi\). For \(0<\theta<\pi\) the directed axis is determined. At identity any axis gives the same zero rotation; at a half-turn both \(n\) and \(-n\) give the same rotation. Outside the principal range, equivalent descriptions include \((n,\theta+2k\pi)\) and \((-n,-\theta)\).
A matrix fixes the direction of this equivalent rotation axis, not its location in physical space. Rotating a club about different parallel lines produces the same orientation change and different translations. A full rigid-body displacement needs position as well as orientation; see the screw-theory reference.
Exponential Coordinates (Lie Algebra \(\mathfrak{so}(3)\))
The rotation vector \(\rho=\theta n\) has three real components; its skew matrix belongs to the Lie algebra:
\[\rho\in\mathbb R^3,\qquad [\rho]_\times\in\mathfrak{so}(3),\qquad R=\exp([\rho]_\times).\]
Where a selected logarithm branch is defined, \([\rho]_\times=\log R\) and \(\rho=(\log R)^\vee\), with vee the inverse of the skew map. A vector is not literally a \(3\times3\) matrix logarithm. The principal vector cannot describe accumulated revolutions: 0 and \(2\pi n\) exponentiate to the same orientation.
For a differentiable orientation history,
\[[\omega_s]_\times=\dot R R^T,\qquad [\omega_b]_\times=R^T\dot R,\qquad \omega_s=R\omega_b.\]
Angular velocity has units rad/s; a rotation vector uses radians. In a smooth rotation-vector chart,
\[\omega_s=J_l(\rho)\dot\rho,\qquad \omega_b=J_l(-\rho)\dot\rho,\]
\[J_l(\rho)=I+\frac{1-\cos\theta}{\theta^2}[\rho]_\times+ \frac{\theta-\sin\theta}{\theta^3}[\rho]_\times^2,\qquad \theta=\|\rho\|.\]
At zero use coefficient limits \(1/2\) and \(1/6\). For fixed-axis motion \(\rho=\theta(t)n\), the cross terms annihilate \(\dot\rho\), giving \(\omega=\dot\theta n\). For a changing axis, simply differentiating rotation-vector components generally gives the wrong angular velocity. Crossing a principal-logarithm branch requires a continuity strategy; exponential coordinates do not provide a global singularity-free three-parameter chart.
Quaternions
A unit Hamilton quaternion is
\[q=(w,v)=\left(\cos\frac\theta2,\ n\sin\frac\theta2\right),\qquad w^2+v^Tv=1.\]
Quaternion to Rotation Matrix
\[R=\begin{bmatrix} 1-2(y^2+z^2)&2(xy-wz)&2(xz+wy)\\ 2(xy+wz)&1-2(x^2+z^2)&2(yz-wx)\\ 2(xz-wy)&2(yz+wx)&1-2(x^2+y^2) \end{bmatrix}.\]
Normalize a finite nonzero quaternion first. Rescaling by its largest component before taking the norm prevents avoidable floating-point overflow or underflow.
Quaternion Composition
For \(p=(p_w,p_v)\) and \(q=(q_w,q_v)\),
\[p\otimes q=(p_wq_w-p_v^Tq_v,\ p_wq_v+q_wp_v+p_v\times q_v).\]
Then \(R(p\otimes q)=R(p)R(q)\), so \(q\) acts first. For unit \(q\), the conjugate \((w,-v)\) represents the inverse. Scalar-last libraries require reordering; changing component order does not change the declared multiplication convention automatically.
Quaternion Derivative
For body-to-world orientation, the Hamilton convention gives
\[\dot q=\tfrac12(0,\omega_s)\otimes q =\tfrac12 q\otimes(0,\omega_b).\]
Both velocities describe the same physical rotation in different frames. Mixing one velocity’s components with the other’s multiplication side changes the motion.
\(q\) and \(-q\) describe one orientation. For nearby sampled orientations, negating a sample when its dot product with the previous one is negative selects a locally continuous, shorter quaternion arc. This assumes the physical increment is resolved by the sampling. At a zero dot product the shortest rotation is not unique. Sparse samples do not reveal extra revolutions or the actual path; a shortest-path interpolation is a modeling choice. An integrated continuous quaternion history need not be forced into the half-sphere \(w\geq0\) at every step.
Conversion Summary Table
| Conversion | Stable Procedure and Limits |
|---|---|
| Axis-angle → \(R\) | Normalize a nonzero axis; apply Rodrigues or its equivalent half-angle formula |
| Quaternion → \(R\) | Normalize a nonzero quaternion; use the unit formula above |
| \(R\) → quaternion | Validate \(SO(3)\); recover the largest quaternion component first, then normalize |
| \(R\) → axis-angle | From a quaternion with \(w\geq0\), use \(\theta=2\operatorname{atan2}(\|v\|,w)\); \(n=v/\|v\|\) unless identity |
| \(R\) → ZYX | Use atan2 pitch and the declared generic or gimbal-lock branch |
| Rotation vector → \(R\) | Identity at zero; otherwise normalize its direction and use its norm as angle |
Why the Largest Quaternion Component Matters
For an exact rotation,
\[\begin{bmatrix}w^2\\x^2\\y^2\\z^2\end{bmatrix} =\frac14\begin{bmatrix}1+\operatorname{tr}R\\1+2R_{11}-\operatorname{tr}R\\1+2R_{22}-\operatorname{tr}R\\1+2R_{33}-\operatorname{tr}R\end{bmatrix}.\]
The four squares sum to one, so the largest component has magnitude at least \(1/2\). Choose it positive and recover the other components from identities such as
\[4wx=R_{32}-R_{23},\quad4wy=R_{13}-R_{31},\quad4wz=R_{21}-R_{12},\]
\[4xy=R_{12}+R_{21},\quad4xz=R_{13}+R_{31},\quad4yz=R_{23}+R_{32}.\]
Divide only by four times the selected component, then normalize. The denominator is at least two for an exact rotation. Always dividing by \(4w\) fails near a half-turn. The trace-based angle \(\arccos((\operatorname{tr}R-1)/2)\) and axis formula using \(1/(2\sin\theta)\) are useful derivations, but lose numerical information near zero or \(\pi\); they are not unconditional algorithms.
Python Implementation
Run this example from the AffineDrift repository root with its Python dependencies installed. The small reference adapter reuses the reviewed Volume 0 quaternion implementation. The complete implementation and tests are in src/tools/rotation_reference_examples.py and tests/test_rotation_representations_reference.py.
The Python matrix tolerance is \(10^{-10}\). The browser accepts maximum entry orthogonality and determinant residuals below \(10^{-5}\) to accommodate rounded editable matrices. Those residuals are not confidence intervals. Approximate accepted matrices and normalized output representations may differ by rounding; neither adapter promises a statistically justified or nearest-rotation fit. The numerical lock threshold is \(|\cos\theta|<10^{-12}\).
import numpy as np
from src.tools.rotation_reference_examples import (
axis_angle_to_rotation as axis_angle_to_R,
euler_zyx_to_rotation as euler_zyx_to_R,
quaternion_to_rotation as quaternion_to_R,
rotation_to_axis_angle as R_to_axis_angle,
rotation_to_euler_zyx as R_to_euler_zyx,
rotation_to_quaternion as R_to_quaternion,
)
# --- Verification: compare reconstructed orientations, not ambiguous coordinates.
axis = np.array([-1.0, 2.0, -3.0]) / np.sqrt(14)
for angle in [1e-12, 0.7, np.pi - 1e-8, np.pi, np.pi + 1e-8]:
original = axis_angle_to_R(axis, angle)
recovered_axis, recovered_angle = R_to_axis_angle(original)
recovered_q = R_to_quaternion(original)
np.testing.assert_allclose(
axis_angle_to_R(recovered_axis, recovered_angle), original, atol=2e-14
)
np.testing.assert_allclose(quaternion_to_R(recovered_q), original, atol=2e-14)
for pitch in [np.pi / 2, -np.pi / 2, np.pi / 2 - 1e-8]:
original = euler_zyx_to_R(0.7, pitch, -0.9)
angles = R_to_euler_zyx(original)
np.testing.assert_allclose(euler_zyx_to_R(*angles), original, atol=2e-14)From Orientation to Golf Mechanics
For calibrated segment-to-world matrices \(R_{WA}\) and \(R_{WB}\), relative segment orientation is
\[R_{AB}=R_{WA}^TR_{WB}.\]
This is more precise than subtracting independently extracted yaw angles. With both segment angular velocities expressed in world coordinates,
\[\omega_{B/A}^{B}=R_{WB}^T(\omega_B^W-\omega_A^W).\]
Anatomical axis calibration, measurement times, filtering and Euler convention must be declared before a relative angle is called pelvic rotation, thorax rotation or a joint excursion. A common fixed world-frame rotation cancels from \(R_{AB}\); independent segment-calibration errors do not.
For a clubface normal fixed in the club frame, \(n_W=R_{WC}n_C\) and
\[\dot n_W=\omega_C^W\times n_W.\]
Angular velocity parallel to the normal does not change that normal instantaneously. Thus an orientation component, angular-speed magnitude and the rate of change of a delivery direction answer different questions. A face normal alone also leaves rotation about that normal undetermined.
For small spatial orientation error \(\delta\vartheta\),
\[R_{\mathrm{measured}}\approx\exp([\delta\vartheta]_\times)R_{\mathrm{true}},\qquad \delta n\approx-[n]_\times\delta\vartheta.\]
Its normal covariance is approximately \([n]_\times\Sigma_\vartheta[n]_\times^T\) in that same frame. Shared calibration and synchronization errors create correlations; treating each segment’s uncertainty as independent can misstate relative-motion uncertainty. Differencing successive orientations requires a declared time interval and a local logarithm branch, and amplifies noise as the interval shrinks. An isolated orientation cannot determine angular velocity.
Finally, rotational kinetic energy needs physical central inertia and angular velocity in a common basis:
\[K=\tfrac12m\|v_C\|^2+\tfrac12\omega^TI_C\omega.\]
Moment power is \(\tau\cdot\omega\), with force power at its actual application point added where appropriate. Neither a large rotation angle nor a peak Euler rate demonstrates proximal-to-distal energy transfer. Those arguments require the segment and interface work ledger, contact constraints, continuing actuator work and uncertainty. Representation changes preserve the physical problem when every vector, inertia and derivative is transformed consistently.
Cross-Reference With Book Chapters
| Topic | Published Reading |
|---|---|
| Frames, quaternion algebra, interpolation and rotational sensitivity | Volume 0: Rotations and SE(3) |
| Ordered joint rotations and space/body Jacobians | Volume 0: Product of Exponentials |
| Power duality and physical inertia | Volume 0: Spatial Algebra |
| Physical energy and intersegment work | Physics of Golf: Energy Transfer |
Sources and Scope
Kevin M. Lynch and Frank C. Park, Modern Robotics (2017), provide the rotation, frame and exponential framework. The official rotation-exponential transcript explains the constant-axis construction; the angular-velocity transcript distinguishes body and spatial representations. The numerical branches and verification examples here are specified explicitly rather than attributed wholesale to those transcripts.
SciPy’s Euler conversion documentation explains intrinsic/extrinsic sequences and its singularity convention. Its matrix conversion documentation describes fitting an approximate rotation; that behavior must not be mistaken for the stricter input validation used here. These mathematical and software sources do not validate a golfer-specific anatomical calibration, coaching intervention or energy-transfer claim.
Citation
@online{olson,
author = {Olson, Dieter},
title = {Rotation {Representations:} {A} {Cross-Reference} {Guide}},
date = {},
url = {https://affinedrift.com/articles/rotation-representations-reference.html},
langid = {en}
}