deficit_eq

formal source

  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
 108(hinge, edge) pair. The identity below constrains this matrix. -/
 109
 110/-- A matrix of deficit-angle derivatives: `dThetadL h e` is intended
 111    to be `∂(totalTheta h) / ∂(len e)`. -/
 112structure DeficitDerivativeMatrix (nH nE : ℕ) where
 113  dThetadL : Fin nH → Fin nE → ℝ
 114
 115/-! ## §3. Schläfli's identity as a hypothesis -/
 116
 117/-- **SCHLÄFLI'S IDENTITY** (piecewise-flat form).
 118
 119    For a finite collection of hinges (indexed by `Fin nH`) with areas
 120    `A_h` and a matrix `dThetadL` of dihedral-angle derivatives with
 121    respect to edge lengths, the weighted sum vanishes:
 122
 123    `∀ e, Σ_h A_h · (∂θ_h / ∂L_e) = 0`.
 124
 125    This is the classical local identity; see Regge (1961, eq. 2.8) and
 126    Brewin (2000). We record it as a hypothesis structure because the
 127    full proof requires boundary-integration machinery not yet in
 128    Mathlib. -/
 129def SchlaefliIdentity {nH nE : ℕ}