File size: 2,280 Bytes
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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 | 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
|