pairwise_winding — DRIVE braidpath op

• Data kinds: any → matrix

• Call: import fullseye as fs; fs.ledger.pairwise_winding(trajectories, obstacles=None) (to call the implementation directly, import braidpath; braidpath.pairwise_winding(trajectories, obstacles=None); from the registry, opsdrive.get("pairwise_winding"))

• GPU: none (CPU reference implementation only)

Usage

Relative winding number for each pair of strands [turns] (total change of the argument of the relative vector / 2π). Symmetric matrix, zero diagonal.

The increment of the argument is taken as atan2(cross, dot) (principal value (−π, π]). If a single step turns half a revolution or more, the direction is undetermined, so it stops. Closed form: for a pure braid, with c the signed number of crossings between the two strands, 2π·W = π c exactly (gate). The winding number is a commutative invariant, so it cannot distinguish braids of the commutator form [A, B] (winding number 0 for every pair, yet nontrivial) —— which is why braids are used.

Returns: (n, n) float array (n = agents + obstacle points).

Raises: ValueError (two strands reach the same point, a single-step increment reaches π —— use finer steps).

References (sample data, literature)

• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).

• Operator provenance and references — the sources of the research/methods this op family came from.

• The canonical algorithm (author, year) and its uses are named in the family usage guide above.

Runnable examples (verified samples that actually call this op)

• poc_braid_homotopy_classes — py -3.11 examples/poc_braid_homotopy_classes.py

Ops the type connects to (they accept matrix as input)

world_camera · lidar_scan · relative_motion · foe_from_motion · flow_from_depth_motion · ttc_truth · world_materials · mesh_signed_volume

Same category (braidpath)

braid_from_trajectories · braid_reduce · braid_artin_images · dynnikov_coordinates · dynnikov_act · braid_equivalent · homotopy_class_compare · homotopy_shortest_paths

*Provenance: braidpath.py — DRIVE operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*

© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.