File size: 2,356 Bytes
695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Geometry.RuledCovering.Isometries
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
/-- Scaled-plane approximation under every Euclidean isometry. -/
theorem physicalHyperbolic : PhysicalHyperbolicStatement := by
intro h L D v hh hL hD hv hH e he F
exact (physical_hyperbolic_cover hh hL hD hv hH he).isometry F
lemma norm_plane_sq (x : Plane) : ‖x‖ ^ 2 = x 0 ^ 2 + x 1 ^ 2 := by
simp only [EuclideanSpace.real_norm_sq_eq, Fin.sum_univ_two]
lemma physicalWeight_one (g : Plane) : physicalWeight 1 g = weight g := by
simp only [physicalWeight, weight, norm_plane_sq, one_pow, one_mul]
congr 2
ring
/-- Cyclic permutation from the transverse physical normalization to (p+rv,r). -/
def cycleCoordinates : Space ≃ₗᵢ[ℝ] Space where
toFun z := vec3 (z 1) (z 2) (z 0)
invFun z := vec3 (z 2) (z 0) (z 1)
left_inv z := by ext i; fin_cases i <;> rfl
right_inv z := by ext i; fin_cases i <;> rfl
map_add' z w := by ext i; fin_cases i <;> rfl
map_smul' c z := by ext i; fin_cases i <;> rfl
norm_map' z := by
change ‖vec3 (z 1) (z 2) (z 0)‖ = ‖z‖
have hz : z = vec3 (z 0) (z 1) (z 2) := by ext i; fin_cases i <;> rfl
have hs : ‖z‖ ^ 2 = z 0 ^ 2 + z 1 ^ 2 + z 2 ^ 2 := by
conv_lhs => rw [hz, norm_vec3_sq]
have ht := norm_vec3_sq (z 1) (z 2) (z 0)
nlinarith [norm_nonneg (vec3 (z 1) (z 2) (z 0)), norm_nonneg z]
lemma cycle_physicalRuled (D : Set Plane) (v : Plane → Plane) (L : ℝ) :
cycleCoordinates '' physicalRuled 1 D v L = ruled D v L := by
ext z
constructor
· rintro ⟨_, ⟨p, hp, r, hr, rfl⟩, rfl⟩
exact ⟨p, hp, r, hr, by simp [cycleCoordinates]⟩
· rintro ⟨p, hp, r, hr, rfl⟩
refine ⟨vec3 r (p 0+r*v p 0) (p 1+r*v p 1),
⟨p, hp, r, hr, by simp⟩, ?_⟩
rfl
/-- General hyperbolic approximation: arbitrary compact labels, including
positive-area frontiers and area-zero sets. -/
theorem hyperbolic : HyperbolicStatement := by
intro D v L hD hv hL hH e he
have hc := physicalHyperbolic 1 L D v (by norm_num) hL hD hv hH e he
cycleCoordinates.toIsometryEquiv
simpa only [LinearIsometryEquiv.coe_toIsometryEquiv, cycle_physicalRuled,
physicalWeight_one, one_mul] using hc
end RuledApproximation
end
end OAI
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