| import OAI.Geometry.CylinderCovering.Barycentric.SectorNeighborhoodDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions |
| import OAI.Geometry.CylinderCovering.Angular.CapsDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped BigOperators |
|
|
| namespace RadialCoordinates.BarycentricSector |
| open CylinderCovering.RadialRiemann |
|
|
| def intercepts (k c l r : ℝ) : Set Plane := |
| {z | ∃ t ∈ Icc l r, ∃ S ∈ Icc (0:ℝ) (cap k t), ∃ p ∈ Icc (-1:ℝ) 1, |
| z = S • ray t + (k*p) • (velocity t-velocity c)} |
|
|
| def canonicalCapBound (k : ℝ) : ℝ := 1/2+(501/1000)*k^2 |
|
|
| structure EnclosureStatement : Prop where |
| triangles : ∀ l r a b : ℝ, l ≤ r → 0 ≤ a → 0 ≤ b → |
| IsCompact (signedSector l r a b) ∧ |
| signedSector l r a b = convexHull ℝ (range (vertices l r b)) ∪ |
| (fun z : Plane => -z) '' convexHull ℝ (range (vertices l r a)) ∧ |
| positiveSector l r b ∩ (fun z : Plane => -z) '' positiveSector l r a = {0} |
| exactArea : ∀ l r a b : ℝ, l ≤ r → 0 ≤ a → 0 ≤ b → |
| volume.real (signedSector l r a b) = (r-l)*(b^2+a^2)/2 |
| nondegenerate : ∀ l r A : ℝ, l < r → 0 < A → |
| AffineIndependent ℝ (vertices l r A) |
| uniformPerimeters : ∀ l r k M M₀ : ℝ, |
| 0 ≤ l → l ≤ r → r ≤ 1 → 0 ≤ k → 0 ≤ M → M ≤ M₀ → |
| perimeter l r (2*k*(r-l)) + perimeter l r (M+2*k*(r-l)) ≤ |
| (2+Real.sqrt 2)*(M₀+4*k) |
| actualIntercepts : ∀ l r c S k p t M : ℝ, |
| 0 ≤ k → |p| ≤ 1 → c ∈ Icc l r → t ∈ Icc l r → 0 ≤ S → S ≤ M → |
| S • ray t + (k*p) • (velocity t-velocity c) ∈ neighborhood l r k M |
| geometricError : ∀ l r k M M₀ : ℝ, |
| 0 ≤ l → l ≤ r → r ≤ 1 → 0 ≤ k → 0 ≤ M → M ≤ M₀ → |
| IsCompact (neighborhood l r k M) ∧ |
| 0 ≤ volume.real (neighborhood l r k M)-(r-l)*M^2/2 ∧ |
| volume.real (neighborhood l r k M)-(r-l)*M^2/2 ≤ |
| (18*k*M₀+162*k^2)*(r-l)^2 |
| weightedFinite : ∀ N : ℕ, 0 < N → ∀ k M₀ : ℝ, 0 ≤ k → |
| ∀ M : Fin N → ℝ, (∀ i, 0 ≤ M i) → (∀ i, M i ≤ M₀) → |
| ∀ S : Fin N → Set Plane, |
| (∀ i, S i ⊆ neighborhood (point 0 1 N i) (point 0 1 N (i+1)) k (M i)) → |
| ∀ W : Fin N → ℝ, (∀ i, 0 ≤ W i) → (∀ i, W i ≤ 1) → |
| (∑ i : Fin N, W i * volume.real (S i)) ≤ |
| (∑ i : Fin N, W i * ((1/(N:ℝ))*(M i)^2/2)) + |
| (18*k*M₀+162*k^2)/(N:ℝ) |
| canonicalCompact : ∀ k c l r : ℝ, IsCompact (intercepts k c l r) |
| canonicalFinite : ∀ N : ℕ, 0 < N → ∀ k : ℝ, 0 ≤ k → |
| ∀ tag : Fin N → ℝ, IsTagged 0 1 N tag → |
| ∀ W : Fin N → ℝ, (∀ i, 0 ≤ W i) → (∀ i, W i ≤ 1) → |
| (∑ i : Fin N, W i * volume.real |
| (intercepts k (tag i) (point 0 1 N i) (point 0 1 N (i+1)))) ≤ |
| (∑ i : Fin N, W i*((1/(N:ℝ))*(capMax (cap k) 0 1 N i)^2/2)) + |
| (18*k*canonicalCapBound k+162*k^2)/(N:ℝ) |
|
|
| end RadialCoordinates.BarycentricSector |
|
|
| end |
|
|
| end OAI |
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|
