Transformation Fields¶
research/transformation_fields explores differentiable fields of local
Clifford-generated transformations. The field machinery stays outside clifra's
stable core: sampling, interpolation, objectives, and application structure are
ordinary PyTorch code around planned Clifford actions.
For a bivector generator \(B\),
Equivalently, on the requested representation,
A transformation field makes the generator depend on a sample identity \(\xi\). The value being transformed and the identity selecting its generator need not be the same object:
This distinction lets a material coordinate, acquisition time, or spatial position remain attached to a sample while the transformed value changes. For fixed identities, reversing the ordered actions and negating their generators gives the indexed inverse.
Experiments¶
The examples use the same basic construction in three different geometries.
| Research example | Core primitives used | Role in the experiment |
|---|---|---|
| Sparse-Constraint Continuum Threading | AlgebraContext(4, 1, 0), grade layout, plan_versor_action(grade=2), geometric_product |
Compact conformal points and bivectors; planned actions deform the rod, and a geometric product normalizes points when extracting coordinates. |
| Continuous-Time PGA LiDAR Deskewing | AlgebraContext(3, 0, 1), grade layout, plan_versor_action(grade=2), plan_product(op="wedge") |
Planned motor actions deskew point trivectors; planned wedges construct lines and evaluate point-plane and point-line geometry. |
| Relativistic Beamline Inverse Design | AlgebraContext(1, 3), grade layout, plan_versor_action(grade=2) |
Planned Lorentz rotor actions update four-velocities while preserving the mass shell. |
See Signatures and layouts, Induced actions, and Planned execution for the Core contracts behind these examples.
Sparse-Constraint Continuum Threading¶
A material-coordinate field in \(Cl(4,1)\) is fitted from three gate poses and one tip pose. The predicted centerline supplies gate crossings without a material-to-gate assignment. The field is then evaluated on a much denser continuum discretization without retraining.
Continuous-Time PGA LiDAR Deskewing¶
A motor field in \(Cl(3,0,1)\) uses acquisition time as persistent identity. Sparse point-plane, point-line, and anchor constraints recover continuous ego-motion and deskew an independently sampled scan.
Relativistic Beamline Inverse Design¶
A \(Cl(1,3)\) electromagnetic bivector field steers a relativistic particle bunch through offset apertures toward a target. Lorentz rotor updates preserve the four-velocity mass shell while the field is optimized end to end.