cube_dihedral

formal source

 102    exact lt_of_le_of_ne h_le h_ne
 103
 104/-- A cube's dihedral angle has cosine `0` (i.e. 90°). -/
 105def cube_dihedral : DihedralAngleData :=
 106  { cosine := 0
 107  , cosine_lb := by norm_num
 108  , cosine_ub := by norm_num
 109  }
 110
 111/-- The cube dihedral is `π / 2` exactly. -/
 112theorem cube_dihedral_theta : cube_dihedral.theta = Real.pi / 2 := by
 113  unfold DihedralAngleData.theta cube_dihedral
 114  simp only
 115  exact Real.arccos_zero
 116
 117/-! ## §3. Flat-sum conditions
 118
 119At a hinge (edge in 3D, 2-face in 4D) embedded in flat Euclidean space,
 120the dihedral angles of the simplices meeting at the hinge sum to `2π`.
 121This is the piecewise-flat analog of "no curvature at this hinge."
 122
 123We formalize this as a condition on a `List DihedralAngleData`. -/
 124
 125/-- The sum of the `theta` values over a list of dihedral data. -/
 126def sumThetas (ds : List DihedralAngleData) : ℝ :=
 127  (ds.map DihedralAngleData.theta).sum
 128
 129/-- The flat-sum condition: the total dihedral angle at a hinge equals `2π`. -/
 130def FlatSumCondition (ds : List DihedralAngleData) : Prop :=
 131  sumThetas ds = 2 * Real.pi
 132
 133/-- The deficit at a hinge is `2π − Σ θ`. -/
 134def deficit (ds : List DihedralAngleData) : ℝ :=
 135  2 * Real.pi - sumThetas ds