Rigid-Body Motion & Transforms

$T = \begin{bmatrix} R & t \\ 0 & 1 \end{bmatrix}$, so a point in frame $B$ maps to frame $A$ by $^A p = {}^AT_B \, {}^Bp$ (with $p$ padded with a 1).

Composing frames is now just matrix multiplication: $^AT_C = {}^AT_B \, {}^BT_C$. Inverting is cheap: $T^{-1} = \begin{bmatrix} R^\top & -R^\top t \\ 0 & 1 \end{bmatrix}$.

Rotation by angle $\theta$ about unit axis $\hat{k}$:

$R = I + \sin\theta\,[\hat{k}]_\times + (1-\cos\theta)\,[\hat{k}]_\times^2$

where $[\hat{k}]_\times$ is the skew-symmetric matrix with $[\hat{k}]_\times v = \hat{k} \times v$. This is the bridge from axis-angle to a full rotation matrix.

To rotate a point $p$, write it as a pure quaternion $(0, p)$ and sandwich it: $p' = q\,p\,q^{*}$ (unit $q$ makes $q^{-1} = q^*$).

Composing rotations is quaternion multiplication: $q_{AC} = q_{AB}\,q_{BC}$ — cheaper and more numerically stable to renormalize than a rotation matrix.

For unit quaternion $(w,x,y,z)$:

$R = \begin{bmatrix} 1-2(y^2+z^2) & 2(xy-wz) & 2(xz+wy) \\ 2(xy+wz) & 1-2(x^2+z^2) & 2(yz-wx) \\ 2(xz-wy) & 2(yz+wx) & 1-2(x^2+y^2) \end{bmatrix}$

Used whenever a downstream computation (e.g. transforming a batch of points) wants an explicit matrix instead of quaternion algebra.

$so(3)$ elements are skew-symmetric matrices $[\omega]_\times$; the matrix exponential $\exp([\omega]_\times) = R$ maps an axis-angle vector to a rotation (Rodrigues' formula is exactly this exponential in closed form).

The log map inverts it, recovering axis and angle from $R$. The $SE(3)$ version does the same for a full rigid-body motion (a "twist"/screw axis), giving a minimal, singularity-aware way to represent and interpolate poses.