| import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions |
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| namespace OAI |
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| noncomputable section |
| namespace CylinderCovering.HalfHeightCaps |
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| def alpha (r : ℝ) : ℝ := (1 + r ^ 2) / 4 |
| def beta (r : ℝ) : ℝ := r / 2 |
| def F (r : ℝ) : ℝ := r ^ 2 * (1 + r ^ 2) / 16 |
| def cubicCap (δ r : ℝ) : ℝ := 1 / 2 + δ ^ 2 * F r + 4 * δ ^ 3 |
| def fractionalCap (δ r : ℝ) : ℝ := 1 / 2 + δ ^ 2 * F r + δ ^ (5 / 2 : ℝ) |
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| def Covered (δ : ℝ) (R : ℝ → ℝ) : Prop := |
| ∀ X Y Z : ℝ, |X| ≤ 1 - Z → |Y| ≤ Z → |
| (∃ r u : ℝ, |r| ≤ 1 ∧ 0 ≤ u ∧ u ≤ R r ∧ |
| Y = u * r + δ * X * alpha r ∧ Z = u + δ * X * beta r) ∨ |
| (∃ s v : ℝ, |s| ≤ 1 ∧ 0 ≤ v ∧ v ≤ R s ∧ |
| X = v * s - δ * Y * alpha s ∧ 1 - Z = v - δ * Y * beta s) |
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| def CoverageStatement : Prop := |
| (∀ δ : ℝ, 0 < δ → Covered δ (cubicCap δ)) ∧ |
| (∃ δ₀ : ℝ, 0 < δ₀ ∧ ∀ δ : ℝ, 0 < δ → δ ≤ δ₀ → Covered δ (fractionalCap δ)) ∧ |
| (∀ δ r : ℝ, 2 * cubicCap δ r = RadialCaps.cubicCap 64 (δ / 2) r) ∧ |
| (∀ δ r : ℝ, 0 < δ → 2 * fractionalCap δ r = |
| RadialCaps.fractionalCap (8 * Real.sqrt 2) (δ / 2) r) |
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| def CompensationStatement : Prop := |
| ∀ δ X Y Z r s u v : ℝ, |
| Y = u * r + δ * X * alpha r → Z = u + δ * X * beta r → |
| X = v * s - δ * Y * alpha s → 1 - Z = v - δ * Y * beta s → |
| (u - 1 / 2) * (1 - δ * X * r) + (v - 1 / 2) * (1 + δ * Y * s) = |
| δ ^ 2 * (X ^ 2 * alpha r + Y ^ 2 * alpha s) |
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| def physicalPoint (X Y Z : ℝ) : RadialPairedSweep.Space := |
| RadialPairedSweep.point (X / 2) (Y / 2) (Z / Real.sqrt 2) |
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| def physicalTetrahedron : Set RadialPairedSweep.Space := |
| {z | ∃ X Y Z : ℝ, |X| ≤ 1 - Z ∧ |Y| ≤ Z ∧ z = physicalPoint X Y Z} |
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| def firstPhysicalSweep (δ : ℝ) (R : ℝ → ℝ) : Set RadialPairedSweep.Space := |
| {z | ∃ r ∈ Set.Icc (-1 : ℝ) 1, ∃ u ∈ Set.Icc (0 : ℝ) (R r), |
| ∃ X : ℝ, |X| ≤ 1 ∧ z = physicalPoint X (u * r + δ * X * alpha r) |
| (u + δ * X * beta r)} |
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| def secondPhysicalSweep (δ : ℝ) (R : ℝ → ℝ) : Set RadialPairedSweep.Space := |
| {z | ∃ s ∈ Set.Icc (-1 : ℝ) 1, ∃ v ∈ Set.Icc (0 : ℝ) (R s), |
| ∃ Y : ℝ, |Y| ≤ 1 ∧ z = physicalPoint (v * s - δ * Y * alpha s) Y |
| (1 - v + δ * Y * beta s)} |
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| def PhysicalCoverageStatement : Prop := |
| (∀ δ : ℝ, 0 < δ → physicalTetrahedron ⊆ |
| firstPhysicalSweep δ (cubicCap δ) ∪ secondPhysicalSweep δ (cubicCap δ)) ∧ |
| (∀ δ : ℝ, 0 < δ → δ ≤ 1 → physicalTetrahedron ⊆ |
| firstPhysicalSweep δ (fractionalCap δ) ∪ secondPhysicalSweep δ (fractionalCap δ)) |
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| end CylinderCovering.HalfHeightCaps |
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| end |
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| end OAI |
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