SE(3)
SE(3) is the group of 3D rotations and translations that preserve distances and orientation.
An SE(3) transform is:
Unlike E(3), it excludes reflections.
Composition is:
and the inverse is:
Pairwise relative vectors transform as:
so their norms are invariant and their directions are equivariant.
Why It Matters
- Many molecular structures can be moved or rotated in space without changing their physical identity.
- SE(3)-equivariant models are useful when predictions include vectors or coordinates.
- The distinction between SE(3), E(3), and SO(3) affects what symmetries are enforced.
Output Contracts
- Scalar: .
- Vector field: under row-vector convention.
- Coordinates: .
State the convention so equations and code agree.
Failure Modes
- Centering removes translation but hides whether deployment uses the same centering rule.
- Reflection-sensitive chemistry is modeled with E(3) when SE(3) is intended.
- Vector labels such as forces are not transformed together with augmented coordinates.
- A bound ligand pose defines the coordinate frame for a task where the ligand pose is unknown at inference time.
Checks
- Are translations and rotations both irrelevant to the input coordinate frame?
- Should reflections be excluded because orientation or chirality matters?
- Are outputs invariant scalars or SE(3)-equivariant vectors/coordinates?
- Does centering or alignment accidentally remove part of the modeling problem?
- Are coordinate units and row/column vector conventions explicit?
- Does preprocessing use only deployment-available information?