| import OAI.Geometry.CylinderCovering.Radial.CapsDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.SweepDefinitions |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory Filter |
| open scoped InnerProductSpace Topology |
| noncomputable section |
|
|
| namespace CylinderCovering.RadialPairedSweep |
|
|
| abbrev Space := TriangularCovering.Space |
|
|
| def point (x y z : ℝ) : Space := !₂[x, y, z] |
|
|
| def tetrahedron (h : ℝ) : Set Space := |
| {z | ∃ x y t : ℝ, |x| ≤ 1 + t ∧ |y| ≤ 1 - t ∧ z = point x y (h * t)} |
|
|
| def upperSweep (h ε : ℝ) (R : ℝ → ℝ) : Set Space := |
| {z | ∃ q ∈ Icc (-1 : ℝ) 1, ∃ ρ ∈ Icc (0 : ℝ) (R q), |
| ∃ x : ℝ, |x| ≤ 2 ∧ |
| z = point x (ρ * q - ε * x * RadialCaps.G q) (h * (1 - ρ + ε * x * q))} |
|
|
| def lowerSweep (h ε : ℝ) (R : ℝ → ℝ) : Set Space := |
| {z | ∃ q ∈ Icc (-1 : ℝ) 1, ∃ ρ ∈ Icc (0 : ℝ) (R q), |
| ∃ y : ℝ, |y| ≤ 2 ∧ |
| z = point (ρ * q + ε * y * RadialCaps.G q) y (h * (-1 + ρ + ε * y * q))} |
|
|
| def upperAxis (h ε q : ℝ) : Space := |
| let v := point 1 (-ε * RadialCaps.G q) (h * ε * q) |
| ‖v‖⁻¹ • v |
|
|
| def lowerAxis (h ε q : ℝ) : Space := |
| let v := point (ε * RadialCaps.G q) 1 (h * ε * q) |
| ‖v‖⁻¹ • v |
|
|
| def physicalCost (h ε : ℝ) (R : ℝ → ℝ) : ℝ := |
| ∫ q in (-1 : ℝ)..1, |
| R q ^ 2 / (2 * Real.sqrt (1 + ε ^ 2 * (RadialCaps.G q ^ 2 + h ^ 2 * q ^ 2))) |
|
|
| def PairedStatement : Prop := |
| ∀ h ε : ℝ, 0 < h → ∀ R : ℝ → ℝ, |
| ContinuousOn R (Icc (-1 : ℝ) 1) → (∀ q ∈ Icc (-1 : ℝ) 1, 0 ≤ R q) → |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, RadialRiemann.IsTagged (-1) 1 N (tagU N)) → |
| (∀ N, RadialRiemann.IsTagged (-1) 1 N (tagL N)) → |
| ∃ U L : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, |
| (∀ N, 0 < N → |
| (∀ i, (U N i).axis = upperAxis h ε (tagU N i) ∧ |
| (L N i).axis = lowerAxis h ε (tagL N i)) ∧ |
| (∀ z ∈ upperSweep h ε R, ∃ i : Fin N, z ∈ (U N i).carrier) ∧ |
| (∀ z ∈ lowerSweep h ε R, ∃ i : Fin N, z ∈ (L N i).carrier)) ∧ |
| Tendsto (fun N => TriangularCovering.totalArea (U N) + |
| TriangularCovering.totalArea (L N)) atTop (𝓝 (2 * h * physicalCost h ε R)) |
|
|
| end CylinderCovering.RadialPairedSweep |
|
|
| end |
|
|
| end OAI |
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|
