File size: 3,268 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 | import OAI.Geometry.CylinderCovering.Physical.CostDefinitions
import OAI.Geometry.CylinderCovering.Convexity.SimilarityDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.UnitEdgeFootprints
def linearReflection (z : Space) : Space := !₂[-z 1, z 0, -z 2]
def reflection (z : Space) : Space := !₂[-z 1, z 0, height - z 2]
def secondRepresentedPoint (η s ρ y : ℝ) : Space :=
!₂[s * ρ - η * y * (1 / 4 + s ^ 2), y, height * (1 - ρ + 2 * η * y * s)]
def secondSweep (η : ℝ) : Set Space :=
{z | ∃ s ∈ Set.Icc (-1 / 2 : ℝ) (1 / 2), ∃ ρ ∈ Set.Icc (0 : ℝ) (sourceCap η s),
∃ y : ℝ, |y| ≤ 1 / 2 ∧ z = secondRepresentedPoint η s ρ y}
def secondAxis (η m : ℝ) : Space := linearReflection (axis η m)
def secondBase (η d M m : ℝ) : Set Space :=
TriangularCovering.projection (secondAxis η m) '' (reflection '' foot η d M m)
def PairedFamilyStatement : Prop :=
∀ η : ℝ, 0 < η →
∃ C₁ C₂ : (N : ℕ) → Fin N → TriangularCovering.TriCylinder,
(∀ N (i : Fin N),
(C₁ N i).axis = axis η (midpoint N i) ∧
(C₁ N i).base = base η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧
(C₂ N i).axis = secondAxis η (midpoint N i) ∧
(C₂ N i).base = secondBase η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧
TriangularCovering.area (C₁ N i).base =
areaFormula η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧
TriangularCovering.area (C₂ N i).base =
areaFormula η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i)) ∧
(∀ N, 0 < N → ∀ z ∈ sweep η (sourceCap η) ∪ secondSweep η,
∃ i : Fin N, z ∈ (C₁ N i).carrier ∨ z ∈ (C₂ N i).carrier) ∧
Filter.Tendsto (fun N => TriangularCovering.totalArea (C₁ N) +
TriangularCovering.totalArea (C₂ N)) Filter.atTop
(nhds (2 * limitingArea η (sourceCap η)))
def tetrahedron : Set Space :=
{z | ∃ t ∈ Set.Icc (0 : ℝ) 1, z 2 = height * t ∧
|z 0| ≤ (1 - t) / 2 ∧ |z 1| ≤ t / 2}
def CoverageStatement : Prop :=
∃ δ : ℝ, 0 < δ ∧ ∀ η : ℝ, 0 < η → η < δ →
tetrahedron ⊆ sweep η (sourceCap η) ∪ secondSweep η
def FiniteStrictStatement : Prop :=
∃ δ : ℝ, 0 < δ ∧ ∀ η : ℝ, 0 < η → η < δ →
∃ N : ℕ, 0 < N ∧ ∃ C₁ C₂ : Fin N → TriangularCovering.TriCylinder,
(∀ i, (C₁ i).axis = axis η (midpoint N i) ∧
(C₁ i).base = base η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i) ∧
(C₂ i).axis = secondAxis η (midpoint N i) ∧
(C₂ i).base = secondBase η (mesh N) (maximumCap (sourceCap η) N i) (midpoint N i)) ∧
(∀ z ∈ tetrahedron, ∃ i : Fin N, z ∈ (C₁ i).carrier ∨ z ∈ (C₂ i).carrier) ∧
TriangularCovering.totalArea C₁ + TriangularCovering.totalArea C₂ < height / 4
def FullStatement : Prop :=
LocalRealizationStatement ∧ FamilyRealizationStatement ∧ PairedFamilyStatement ∧
SourceCostStatement ∧ CoverageStatement ∧ FiniteStrictStatement ∧
Similarity.minimumArea tetrahedron = height / 2
end CylinderCovering.UnitEdgeFootprints
end
end OAI
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