SimplicialEdgeData

formal source

  74
  75/-- Abstract edge-length data for a simplicial complex with finitely many
  76    edges indexed by `Fin nE`. -/
  77structure SimplicialEdgeData (nE : ℕ) where
  78  len : Fin nE → ℝ
  79  len_pos : ∀ e, 0 < len e
  80
  81/-- Abstract hinge data: each hinge knows its area and the collection of
  82    dihedral angles of the top simplices meeting it. The *total* dihedral
  83    at the hinge is `Σ θ_σ`; the deficit is `2π − Σ θ_σ`. -/
  84structure SimplicialHingeData where
  85  area : ℝ
  86  area_nonneg : 0 ≤ area
  87  dihedrals : List DihedralAngleData
  88
  89namespace SimplicialHingeData
  90
  91/-- Sum of the dihedral angles at the hinge. -/
  92def totalTheta (h : SimplicialHingeData) : ℝ :=
  93  DihedralAngle.sumThetas h.dihedrals
  94
  95/-- Deficit at the hinge: `2π − Σ θ`. -/
  96def deficit (h : SimplicialHingeData) : ℝ :=
  97  DihedralAngle.deficit h.dihedrals
  98
  99theorem deficit_eq (h : SimplicialHingeData) :
 100    h.deficit = 2 * Real.pi - h.totalTheta := rfl
 101
 102end SimplicialHingeData
 103
 104/-! ## §2. Variational data
 105
 106For Schläfli's identity we need derivatives of `θ_h` with respect to each
 107edge length `L_e`. We package these as a matrix of real numbers, one per