SO(3)

SO(3) is the group of 3D rotations around the origin.

The group is:

S O (3) = {R \in R^{3 \times 3} ∣ R^{⊤} R = I, det (R) = 1}

It contains rotations but not translations or reflections.

The group action on a point or vector is:

x^{'} = R x

Composition and inverse are:

R_{3} = R_{2} R_{1}, R^{- 1} = R^{⊤}

Distances to the origin and angles are preserved:

∥ R x ∥_{2} = ∥ x ∥_{2}, (R u)^{⊤} (R v) = u^{⊤} v

Relation to Features

Scalar features are invariant under SO(3).
Vector features transform as $v^{'} = R v$ .
Higher-order tensor features transform under higher-dimensional representations.
Spherical harmonics and irreducible representations organize angular information by rotation behavior.

Rotational symmetry is central to models over coordinates, directions, and local frames.
Spherical features and tensor features are often organized by how they transform under SO(3).
SO(3) is a building block for SE(3) methods that also handle translation.

Using SO(3) alone when translation is also arbitrary.
Treating reflections as rotations; reflections have determinant $- 1$ and are not in SO(3).
Aligning every sample to a canonical frame and accidentally injecting target-specific information.