Core 12 min 2 of 3

Four ways to write a rotation

Matrices, axis–angle, quaternions and Euler angles — each optimal for something, none for everything.

What this repairs

The belief that quaternions are "better" and Euler angles are "bad", rather than each being adapted to a different job.

Assumes

Frames, and a notation that catches mistakes

A rotation in 3D has three degrees of freedom. Every representation is a compromise between that number and the operations you want to be cheap.

	Numbers	Constraint	Good at	Bad at
Matrix $R$	9	$R^\top R = I$ , $\det R = 1$	applying to vectors, composing	storage, staying valid under drift
Axis–angle $\boldsymbol{\omega}$	3	none	minimal, natural for increments	composing, singular at $2\pi$
Quaternion $q$	4	$\lVert q \rVert = 1$	composing, interpolating, stability	reading by eye, sign ambiguity
Euler $(\psi, \theta, \phi)$	3	none	human interfaces	everything else

The three-number representations cannot be globally free of singularities — this is a topological fact about $SO(3)$ , not a failure of imagination — which is why the well-behaved representations carry a redundant number and a constraint.

Pitfall Euler angles are ambiguous until you name six things

A triple of angles means nothing without: the three axes, their order, whether the rotations are intrinsic (about the moving frame) or extrinsic (about the fixed frame), and the handedness. That is $12$ intrinsic conventions and $12$ extrinsic ones. "Roll, pitch, yaw" usually means intrinsic Z-Y-X, but "usually" is not a specification, and three.js, ROS, and most aerospace texts each default to something different.

Then there is gimbal lock: at pitch $= \pm90°$ , yaw and roll rotate about the same physical axis and only their sum is observable. The orientation is perfectly fine; the coordinates are degenerate. A controller differentiating Euler angles through that configuration produces an unbounded command from a bounded motion.

Axis–angle to matrix (Rodrigues)

$\mathbf{v}_{\parallel} = (\mathbf{k}\cdot\mathbf{v})\,\mathbf{k}, \qquad \mathbf{v}_{\perp} = \mathbf{v} - \mathbf{v}_{\parallel}$

Split the vector into components along and across the rotation axis k. Only the perpendicular part moves.
$\mathbf{v}_{\perp}' = \cos\theta\,\mathbf{v}_{\perp} + \sin\theta\,(\mathbf{k}\times\mathbf{v})$

The perpendicular part rotates in its plane, with k × v supplying the perpendicular direction of unit length.
$R = I + \sin\theta\,[\mathbf{k}]_\times + (1-\cos\theta)\,[\mathbf{k}]_\times^2$

Collect terms and write the cross product as the skew-symmetric matrix [k]ₓ. Three numbers in, a valid rotation matrix out — always, with no constraint to enforce.

Check yourself

You store orientations as $3\times3$ matrices and integrate small updates over thousands of steps. What goes wrong, and what fixes it?

Show answer

Floating-point error accumulates and $R$ drifts away from orthogonality — it starts scaling and shearing slightly. Symptoms: vectors changing length under "rotation", and $R^\top$ no longer being the inverse. Fixes, in increasing order of principle: periodic re-orthonormalisation (Gram–Schmidt or the SVD projection $R \leftarrow UV^\top$ ); storing a quaternion and re-normalising, which is one constraint instead of six; or composing updates as $R \leftarrow R\,\exp([\boldsymbol{\omega}]_\times)$ , where the exponential map produces an exactly-orthogonal factor by construction.