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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 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Geometry.CylinderCovering.Physical.FiniteDefinitions
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace RadialPhysical.ExplicitTriangles
open CylinderCovering
def error (centered : Bool) (b eta : ℝ) : ℝ := axialBound centered b * |eta|
def left (centered : Bool) (b eta : ℝ) (N : ℕ) : ℝ :=
-(3 * error centered b eta + 1) * RadialRiemann.step (-b) b N
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
def middle (b : ℝ) (N : ℕ) (i : Fin N) : ℝ :=
(RadialRiemann.point (-b) b N i + RadialRiemann.point (-b) b N (i + 1)) / 2
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))]
def upperIntercept (b : ℝ) (p : ℝ × ℝ) : Space := point b 0 p.2 (1-p.1)
def lowerIntercept (centered : Bool) (b : ℝ) (p : ℝ × ℝ) : Space :=
point b p.2 0 ((if centered then (-1:ℝ) else 0)+p.1)
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
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
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))
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))
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)))
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))
end RadialPhysical.ExplicitTriangles
end
end OAI
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