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785c727 | 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 | import OAI.Geometry.CylinderCovering.Ruled.CellsDefinitions
import OAI.Geometry.RuledCovering.Model
namespace OAI
noncomputable section
namespace CylinderCovering.ProductLocal
open MeasureTheory RuledApproximation
def affineCell (p : Plane) (rho : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane)
(S : Set (ℝ × ℝ)) : Set Plane := (fun s => p + rho • L s) '' S
def IsUnitEigenframe (A : Plane →L[ℝ] Plane) (lam : ℝ)
(L : (ℝ × ℝ) ≃L[ℝ] Plane) : Prop :=
‖L (1,0)‖ = 1 ∧ ‖L (0,1)‖ = 1 ∧
A (L (1,0)) = lam • L (1,0) ∧
A (L (0,1)) = (-lam) • L (0,1)
def LocalStatement : Prop :=
∀ (D : Set Plane) (v : Plane → Plane) (h M : ℝ),
IsCompact D → D.Nonempty → 0 < h → 0 < M → SmoothNear 1 D v →
(∀ p ∈ D, HyperbolicAt M v p) →
∀ (eta tau gamma : ℝ), 0 < eta → 0 < tau → tau < 1 → 0 < gamma →
(1+gamma)*tau < 1 → ProductCells.areaRatio tau gamma ≤ 1+eta →
∃ rho0 : ℝ, 0 < rho0 ∧ ∀ p ∈ D,
∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane),
IsUnitEigenframe (fderiv ℝ v p) lam L → 0 < M*lam → M*lam < 1 →
∀ rho : ℝ, 0 < rho → rho < rho0 →
affineCell p rho L (ProductCells.parameterCell tau) ⊆ D →
∃ C : Cylinder,
C.axis = ‖vec3 1 (v p 0) (h*v p 1)‖⁻¹ • vec3 1 (v p 0) (h*v p 1) ∧
C.carrier = firstPhysicalMap h '' auxiliaryCylinder
(affineCell p rho L (ProductCells.interceptCell tau gamma)) (v p) ∧
C.openBounded ∧
physicalRuled h (affineCell p rho L (ProductCells.parameterCell tau)) v M ⊆ C.carrier ∧
C.area ≤ h*(1+eta)^2 *
(∫ q in affineCell p rho L (ProductCells.parameterCell tau), physicalWeight h (v q))
end CylinderCovering.ProductLocal
end
end OAI
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