| 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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|
