totalTheta

formal source

  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
 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