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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 | import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.HalfHeightCaps
def alpha (r : ℝ) : ℝ := (1 + r ^ 2) / 4
def beta (r : ℝ) : ℝ := r / 2
def F (r : ℝ) : ℝ := r ^ 2 * (1 + r ^ 2) / 16
def cubicCap (δ r : ℝ) : ℝ := 1 / 2 + δ ^ 2 * F r + 4 * δ ^ 3
def fractionalCap (δ r : ℝ) : ℝ := 1 / 2 + δ ^ 2 * F r + δ ^ (5 / 2 : ℝ)
def Covered (δ : ℝ) (R : ℝ → ℝ) : Prop :=
∀ X Y Z : ℝ, |X| ≤ 1 - Z → |Y| ≤ Z →
(∃ r u : ℝ, |r| ≤ 1 ∧ 0 ≤ u ∧ u ≤ R r ∧
Y = u * r + δ * X * alpha r ∧ Z = u + δ * X * beta r) ∨
(∃ s v : ℝ, |s| ≤ 1 ∧ 0 ≤ v ∧ v ≤ R s ∧
X = v * s - δ * Y * alpha s ∧ 1 - Z = v - δ * Y * beta s)
def CoverageStatement : Prop :=
(∀ δ : ℝ, 0 < δ → Covered δ (cubicCap δ)) ∧
(∃ δ₀ : ℝ, 0 < δ₀ ∧ ∀ δ : ℝ, 0 < δ → δ ≤ δ₀ → Covered δ (fractionalCap δ)) ∧
(∀ δ r : ℝ, 2 * cubicCap δ r = RadialCaps.cubicCap 64 (δ / 2) r) ∧
(∀ δ r : ℝ, 0 < δ → 2 * fractionalCap δ r =
RadialCaps.fractionalCap (8 * Real.sqrt 2) (δ / 2) r)
def CompensationStatement : Prop :=
∀ δ X Y Z r s u v : ℝ,
Y = u * r + δ * X * alpha r → Z = u + δ * X * beta r →
X = v * s - δ * Y * alpha s → 1 - Z = v - δ * Y * beta s →
(u - 1 / 2) * (1 - δ * X * r) + (v - 1 / 2) * (1 + δ * Y * s) =
δ ^ 2 * (X ^ 2 * alpha r + Y ^ 2 * alpha s)
def physicalPoint (X Y Z : ℝ) : RadialPairedSweep.Space :=
RadialPairedSweep.point (X / 2) (Y / 2) (Z / Real.sqrt 2)
def physicalTetrahedron : Set RadialPairedSweep.Space :=
{z | ∃ X Y Z : ℝ, |X| ≤ 1 - Z ∧ |Y| ≤ Z ∧ z = physicalPoint X Y Z}
def firstPhysicalSweep (δ : ℝ) (R : ℝ → ℝ) : Set RadialPairedSweep.Space :=
{z | ∃ r ∈ Set.Icc (-1 : ℝ) 1, ∃ u ∈ Set.Icc (0 : ℝ) (R r),
∃ X : ℝ, |X| ≤ 1 ∧ z = physicalPoint X (u * r + δ * X * alpha r)
(u + δ * X * beta r)}
def secondPhysicalSweep (δ : ℝ) (R : ℝ → ℝ) : Set RadialPairedSweep.Space :=
{z | ∃ s ∈ Set.Icc (-1 : ℝ) 1, ∃ v ∈ Set.Icc (0 : ℝ) (R s),
∃ Y : ℝ, |Y| ≤ 1 ∧ z = physicalPoint (v * s - δ * Y * alpha s) Y
(1 - v + δ * Y * beta s)}
def PhysicalCoverageStatement : Prop :=
(∀ δ : ℝ, 0 < δ → physicalTetrahedron ⊆
firstPhysicalSweep δ (cubicCap δ) ∪ secondPhysicalSweep δ (cubicCap δ)) ∧
(∀ δ : ℝ, 0 < δ → δ ≤ 1 → physicalTetrahedron ⊆
firstPhysicalSweep δ (fractionalCap δ) ∪ secondPhysicalSweep δ (fractionalCap δ))
end CylinderCovering.HalfHeightCaps
end
end OAI
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