| 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 | |