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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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Geometry.CylinderCovering.Footprints.FootprintsDefinitions
import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions
namespace OAI
open Set Filter MeasureTheory
open scoped Topology InnerProductSpace
noncomputable section
namespace CylinderCovering.HalfheightFinite
abbrev Space := TriangularCovering.Space
def height : ℝ := 1 / Real.sqrt 2
def point (X Y Z : ℝ) : Space := !₂[X/2,Y/2,height*Z]
def tetrahedron : Set Space :=
{z | ∃ X Y Z : ℝ, |X| ≤ 1-Z ∧ |Y| ≤ Z ∧ z = point X Y Z}
def alpha (r : ℝ) : ℝ := (1+r^2)/4
def beta (r : ℝ) : ℝ := r/2
def profile (r : ℝ) : ℝ := r^2*(1+r^2)/16
def cubicCap (δ r : ℝ) : ℝ := 1/2+δ^2*profile r+4*δ^3
def fractionalCap (δ r : ℝ) : ℝ := 1/2+δ^2*profile r+δ^(5/2:ℝ)
def firstRaw (δ r : ℝ) : Space := !₂[1,δ*alpha r,2*height*δ*beta r]
def secondRaw (δ r : ℝ) : Space := !₂[-δ*alpha r,1,2*height*δ*beta r]
def firstAxis (δ r : ℝ) : Space := ‖firstRaw δ r‖⁻¹ • firstRaw δ r
def secondAxis (δ r : ℝ) : Space := ‖secondRaw δ r‖⁻¹ • secondRaw δ r
def firstFoot (p : ℝ × ℝ) : Space := !₂[0,p.2/2,height*p.1]
def secondFoot (p : ℝ × ℝ) : Space := !₂[p.2/2,0,height*(1-p.1)]
def halfwidth (N : ℕ) : ℝ := 1/(N:ℝ)
def midpoint (N : ℕ) (i : Fin N) : ℝ :=
(RadialRiemann.point (-1) 1 N i+RadialRiemann.point (-1) 1 N (i+1))/2
def maxCap (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : ℝ := RadialRiemann.capMax R (-1) 1 N i
def collar (δ : ℝ) (N : ℕ) : ℝ := δ*halfwidth N/2
def foot (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : Set (ℝ × ℝ) :=
RadialFootprints.halfheightFootprint (halfwidth N) (collar δ N) (maxCap R N i) (midpoint N i)
def firstBase (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : Set Space :=
TriangularCovering.projection (firstAxis δ (midpoint N i)) '' (firstFoot '' foot δ R N i)
def secondBase (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : Set Space :=
TriangularCovering.projection (secondAxis δ (midpoint N i)) '' (secondFoot '' foot δ R N i)
def localArea (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : ℝ :=
let M := maxCap R N i
let e := collar δ N
height/2 * halfwidth N * (M^2+6*e*M+8*e^2) /
Real.sqrt (1+δ^2*(alpha (midpoint N i)^2+2*beta (midpoint N i)^2))
def totalArea (δ : ℝ) (R : ℝ → ℝ) (N : ℕ) : ℝ :=
(∑ i : Fin N, TriangularCovering.area (firstBase δ R N i))+
∑ i : Fin N, TriangularCovering.area (secondBase δ R N i)
def limitingArea (δ : ℝ) (R : ℝ → ℝ) : ℝ :=
height/2 * ∫ r in (-1:ℝ)..1, R r^2 /
Real.sqrt (1+δ^2*(alpha r^2+2*beta r^2))
def BaseProperties (u : Space) (B : Set Space) : Prop :=
‖u‖=1 ∧ IsCompact B ∧ Convex ℝ B ∧ MeasurableSet B ∧
Measure.euclideanHausdorffMeasure 2 B ≠ ⊤ ∧ ∀ z ∈ B, inner ℝ u z=0
def FamilyConclusion (δ : ℝ) (R : ℝ → ℝ) : Prop :=
(∀ N : ℕ, 0<N →
(∀ i : Fin N,
BaseProperties (firstAxis δ (midpoint N i)) (firstBase δ R N i) ∧
BaseProperties (secondAxis δ (midpoint N i)) (secondBase δ R N i) ∧
TriangularCovering.area (firstBase δ R N i)=localArea δ R N i ∧
TriangularCovering.area (secondBase δ R N i)=localArea δ R N i) ∧
(∀ z ∈ tetrahedron,
(∃ i : Fin N, z ∈ RadialSweep.cylinder (firstAxis δ (midpoint N i)) (firstBase δ R N i)) ∨
(∃ i : Fin N, z ∈ RadialSweep.cylinder (secondAxis δ (midpoint N i)) (secondBase δ R N i)))) ∧
Tendsto (totalArea δ R) atTop (𝓝 (limitingArea δ R))
def Statement : Prop :=
(∀ δ : ℝ, 0<δ → δ≤1/2 → FamilyConclusion δ (cubicCap δ)) ∧
(∀ᶠ δ : ℝ in 𝓝[>] 0, FamilyConclusion δ (fractionalCap δ)) ∧
((fun δ => limitingArea δ (cubicCap δ)-
height/2*(1/2-δ^2/240+8*δ^3)) =O[𝓝[>] (0:ℝ)] (fun δ => δ^4)) ∧
((fun δ => limitingArea δ (fractionalCap δ)-
height/2*(1/2-δ^2/240+2*δ^(5/2:ℝ))) =O[𝓝[>] (0:ℝ)] (fun δ => δ^4)) ∧
(∀ᶠ δ : ℝ in 𝓝[>] 0,
(∀ᶠ N in atTop, 0<N ∧ totalArea δ (cubicCap δ) N < height/4) ∧
(∀ᶠ N in atTop, 0<N ∧ totalArea δ (fractionalCap δ) N < height/4))
end CylinderCovering.HalfheightFinite
end
end OAI
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