HingeRotation

formal source

  43
  44    For 3D: the hinge is an edge, and the rotation is in the plane
  45    perpendicular to that edge. The rotation angle equals the deficit. -/
  46structure HingeRotation where
  47  axis : Fin 3
  48  angle : ℝ
  49
  50/-- The composition of two rotations about the same axis. -/
  51def compose_same_axis (r1 r2 : HingeRotation) (h : r1.axis = r2.axis) :
  52    HingeRotation :=
  53  ⟨r1.axis, r1.angle + r2.angle⟩
  54
  55/-- A path of rotations around a loop of hinges. -/
  56def loop_holonomy (rotations : List HingeRotation) : ℝ :=
  57  (rotations.map HingeRotation.angle).sum
  58
  59/-! ## The Discrete Bianchi Identity -/
  60
  61/-- **DISCRETE BIANCHI IDENTITY (Hamber-Kagel)**:
  62    The product of rotation matrices along any null-homotopic path
  63    through the dual lattice is the identity matrix.
  64
  65    In terms of deficit angles: for any closed loop of hinges sharing
  66    a common vertex, the sum of (signed) deficit angles equals zero
  67    modulo 2*pi.
  68
  69    More precisely: for the hinges h_1, ..., h_n forming a closed
  70    path around a vertex v in the dual complex:
  71      sum_{i=1}^n delta_{h_i} = 0  (mod 2*pi)
  72
  73    In the linearized (small-angle) regime, this becomes:
  74      sum_{i=1}^n delta_{h_i} = 0  (exactly)
  75
  76    This is the geometric identity that, combined with the Regge