4D Convex Polytopes — Design Notes

Goal: clear, explicit 4D operations with both H/V representations and simple, reliable conversions. Scale is moderate (≈1e6), so naive enumeration is acceptable and keeps dependencies light.

Representations

• H‑rep: half‑spaces n·x <= c, n ∈ R^4.
• V‑rep: vertices x_i ∈ R^4.
• Poly4 stores both; each side may be empty and is filled on demand (ensure_*).

Conversions

• H→V: enumerate all 4‑tuples of inequalities, solve the equalities, keep feasible points. Deduplicate by metric tolerance.
• V→H: enumerate 4‑tuples of vertices, form the unique hyperplane through them using a 3×4 cofactor‑based nullspace (no SVD). Keep only supporting planes (all vertices on one side); orient as n·x <= c.

These are O(N^4) but used infrequently; they keep the implementation compact and transparent.

Faces

• Derive from H‑rep via vertex saturation:
  • 3‑faces (facets): vertices saturating a single inequality.
  • 2‑faces: vertices saturating a pair.
  • 1‑faces (edges): vertices saturating a triple.
  • 0‑faces: the vertices themselves.
• Return simple structs with facet indices and the corresponding vertex list. For downstream geometry we often only need vertices; equalities are kept as indices for traceability.

Symplectic Helpers

• J‑matrix in 4D: J = [[0, -I],[I, 0]].
• Symplectic check: M^T J M ≈ J (tolerance 1e-8).
• Reeb flow on 3‑faces: R_i = J n_i (unnormalized). 1‑faces: stub until the derivation is written in the thesis.

2‑Face → 2D Mapping

• Given two facet indices (i,j), the 2‑face is the plane orthogonal to span{n_i,n_j}.
• Build an orthonormal basis (u1,u2) of that plane via Gram–Schmidt in 4D; map x ↦ y = Ux with U ∈ R^{2×4} (inverse on the plane is x = U^T y).
• Orientation: we expose a boolean to pick the sign; a precise orientation convention (e.g., (u1,u2,n_i,n_j) positively oriented) can be fixed later if needed by algorithms. See “Open Questions”.
• Project the face’s vertices and construct a 2D polytope in H‑rep via Poly2::from_points_convex_hull, giving a faithful 2D model of the face.

Affine Maps

• Push‑forward (H & V): algebraic transform on H‑rep and direct transform on vertices, requiring M invertible.
• Inversion: (M,t) ↦ (M^{-1}, -M^{-1}t).

Open Questions / Escalations

• Orientation of 2‑faces “as crossed by the Reeb flow”: likely induced by the ambient symplectic 2‑form; we left a design hook (sign boolean) and will add a proof‑based convention on request.
• Reeb flow on 1‑faces: placeholder stub until the derivation is added.

Conventions

• Tolerance eps = 1e-9 for feasibility/equality; 1e-8 for symplectic check.
• Keep code explicit and small; use math comments to tie back to this page.
• Tests: smoke tests for cubes/simplex; property tests optional.