| import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions |
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| namespace OAI |
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| open Set Filter |
| open scoped Topology |
| noncomputable section |
| namespace RadialPhysical.ExplicitTriangles |
| open CylinderCovering |
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| def error (centered : Bool) (b eta : ℝ) : ℝ := axialBound centered b * |eta| |
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| def left (centered : Bool) (b eta : ℝ) (N : ℕ) : ℝ := |
| -(3 * error centered b eta + 1) * RadialRiemann.step (-b) b N |
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| def right (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) (N : ℕ) (i : Fin N) : ℝ := |
| RadialRiemann.capMax T (-b) b N i + |
| error centered b eta * RadialRiemann.step (-b) b N |
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| def middle (b : ℝ) (N : ℕ) (i : Fin N) : ℝ := |
| (RadialRiemann.point (-b) b N i + RadialRiemann.point (-b) b N (i + 1)) / 2 |
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| def coordinates (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (i : Fin N) : Fin 3 → ℝ × ℝ := |
| let l := left centered b eta N |
| let r := right centered b eta T N i |
| let m := middle b N i |
| let d := RadialRiemann.step (-b) b N |
| ![(l, m*l), (r, m*r-d/2*(r-l)), (r, m*r+d/2*(r-l))] |
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| def upperIntercept (b : ℝ) (p : ℝ × ℝ) : Space := point b 0 p.2 (1-p.1) |
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| def lowerIntercept (centered : Bool) (b : ℝ) (p : ℝ × ℝ) : Space := |
| point b p.2 0 ((if centered then (-1:ℝ) else 0)+p.1) |
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| def upperVertices (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (i : Fin N) : Fin 3 → Space := |
| upperIntercept b ∘ coordinates centered b eta T N i |
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| def lowerVertices (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (i : Fin N) : Fin 3 → Space := |
| lowerIntercept centered b ∘ coordinates centered b eta T N i |
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| def upperBase (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : Set Space := |
| TriangularCovering.projection (upperPhysicalAxis b eta (tag i)) '' |
| convexHull ℝ (Set.range (upperVertices centered b eta T N i)) |
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| def lowerBase (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : Set Space := |
| TriangularCovering.projection (lowerPhysicalAxis b eta (tag i)) '' |
| convexHull ℝ (Set.range (lowerVertices centered b eta T N i)) |
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| def localArea (centered : Bool) (b eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : ℝ := |
| height b * RadialRiemann.step (-b) b N * |
| (RadialRiemann.capMax T (-b) b N i + |
| (4 * error centered b eta + 1) * RadialRiemann.step (-b) b N)^2 / |
| (2 * Real.sqrt (1 + eta^2 * |
| (((tag i)^2+b^2)^2/4 + height b^2*(tag i)^2))) |
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| def ConstructionStatement : Prop := |
| ∀ (centered : Bool) (b eta : ℝ), 0 < b → ∀ T : ℝ → ℝ, |
| ContinuousOn T (Icc (-b) b) → (∀ s ∈ Icc (-b) b, 0 ≤ T s) → |
| ∀ tagU tagL : (N : ℕ) → Fin N → ℝ, |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagU N)) → |
| (∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) → |
| ∃ U L : (N : ℕ) → Fin N → TriangularCovering.TriCylinder, |
| (∀ N i, |
| (U N i).axis = upperPhysicalAxis b eta (tagU N i) ∧ |
| (L N i).axis = lowerPhysicalAxis b eta (tagL N i) ∧ |
| (U N i).base = upperBase centered b eta T N (tagU N) i ∧ |
| (L N i).base = lowerBase centered b eta T N (tagL N) i ∧ |
| TriangularCovering.area (U N i).base = localArea centered b eta T N (tagU N) i ∧ |
| TriangularCovering.area (L N i).base = localArea centered b eta T N (tagL N) i ∧ |
| upperPhysicalBase centered b eta T N (tagU N) i ⊆ (U N i).base ∧ |
| lowerPhysicalBase centered b eta T N (tagL N) i ⊆ (L N i).base) ∧ |
| (∀ N, 0 < N → |
| (∀ z ∈ upperSweepOn centered b eta T (Icc (-b) b), |
| ∃ i : Fin N, z ∈ (U N i).carrier) ∧ |
| (∀ z ∈ lowerSweepOn centered b eta T (Icc (-b) b), |
| ∃ i : Fin N, z ∈ (L N i).carrier)) ∧ |
| Tendsto (fun N => TriangularCovering.totalArea (U N) + |
| TriangularCovering.totalArea (L N)) atTop (𝓝 (scaledCost b eta T)) |
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| end RadialPhysical.ExplicitTriangles |
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| end |
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| end OAI |
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