File size: 3,240 Bytes
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 | import OAI.Geometry.TriangularCovering.Basic
import OAI.Geometry.CylinderCovering.Radial.RiemannDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.UnitEdgeFootprints
abbrev Space := TriangularCovering.Space
def height : ℝ := 1 / Real.sqrt 2
def intercept (u v : ℝ) : Space := !₂[0, v, height * u]
def rawDirection (η m : ℝ) : Space :=
!₂[1, η * (1 / 4 + m ^ 2), 2 * height * η * m]
def axis (η m : ℝ) : Space := ‖rawDirection η m‖⁻¹ • rawDirection η m
def vertices (η d M m : ℝ) : Fin 3 → Space :=
![intercept (-η * d) (m * (-η * d)),
intercept (M + η * d) (m * (M + η * d) - d / 2 * (M + 2 * η * d)),
intercept (M + η * d) (m * (M + η * d) + d / 2 * (M + 2 * η * d))]
def foot (η d M m : ℝ) : Set Space := convexHull ℝ (Set.range (vertices η d M m))
def base (η d M m : ℝ) : Set Space :=
TriangularCovering.projection (axis η m) '' foot η d M m
def representedPoint (η s ρ x : ℝ) : Space :=
!₂[x, s * ρ + η * x * (1 / 4 + s ^ 2), height * (ρ + 2 * η * x * s)]
def areaFormula (η d M m : ℝ) : ℝ :=
height * d * (M + 2 * η * d) ^ 2 /
(2 * Real.sqrt (1 + η ^ 2 * ((1 / 4 + m ^ 2) ^ 2 + 2 * m ^ 2)))
def LocalRealizationStatement : Prop :=
∀ η d M m : ℝ, 0 < η → 0 < d → 0 ≤ M →
∃ C : TriangularCovering.TriCylinder,
C.axis = axis η m ∧ C.base = base η d M m ∧
TriangularCovering.area C.base = areaFormula η d M m ∧
∀ s ρ x : ℝ, |s - m| ≤ d / 2 → 0 ≤ ρ → ρ ≤ M → |x| ≤ 1 / 2 →
representedPoint η s ρ x ∈ C.carrier
def sourceCap (η s : ℝ) : ℝ :=
Real.sqrt (1 / 4 + η ^ 2 * (s ^ 2 / 4 + s ^ 4 + 1 / 960))
def sweep (η : ℝ) (R : ℝ → ℝ) : Set Space :=
{z | ∃ s ∈ Set.Icc (-1 / 2 : ℝ) (1 / 2), ∃ ρ ∈ Set.Icc (0 : ℝ) (R s),
∃ x : ℝ, |x| ≤ 1 / 2 ∧ z = representedPoint η s ρ x}
def mesh (N : ℕ) : ℝ := RadialRiemann.step (-1 / 2) (1 / 2) N
def midpoint (N : ℕ) (i : Fin N) : ℝ :=
(RadialRiemann.point (-1 / 2) (1 / 2) N i +
RadialRiemann.point (-1 / 2) (1 / 2) N (i + 1)) / 2
def maximumCap (R : ℝ → ℝ) (N : ℕ) (i : Fin N) : ℝ :=
RadialRiemann.capMax R (-1 / 2) (1 / 2) N i
def limitingArea (η : ℝ) (R : ℝ → ℝ) : ℝ :=
height * ∫ s in (-1 / 2 : ℝ)..(1 / 2),
R s ^ 2 / (2 * Real.sqrt (1 + η ^ 2 * ((1 / 4 + s ^ 2) ^ 2 + 2 * s ^ 2)))
def FamilyRealizationStatement : Prop :=
∀ η : ℝ, 0 < η → ∀ R : ℝ → ℝ,
ContinuousOn R (Set.Icc (-1 / 2 : ℝ) (1 / 2)) →
(∀ s ∈ Set.Icc (-1 / 2 : ℝ) (1 / 2), 0 ≤ R s) →
∃ C : (N : ℕ) → Fin N → TriangularCovering.TriCylinder,
(∀ N (i : Fin N),
(C N i).axis = axis η (midpoint N i) ∧
(C N i).base = base η (mesh N) (maximumCap R N i) (midpoint N i) ∧
TriangularCovering.area (C N i).base =
areaFormula η (mesh N) (maximumCap R N i) (midpoint N i)) ∧
(∀ N, 0 < N → ∀ z ∈ sweep η R, ∃ i : Fin N, z ∈ (C N i).carrier) ∧
Filter.Tendsto (fun N => TriangularCovering.totalArea (C N)) Filter.atTop
(nhds (limitingArea η R))
end CylinderCovering.UnitEdgeFootprints
end
end OAI
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