| import OAI.Geometry.RuledCovering.Model |
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|
| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
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| namespace CylinderCovering.BeamStrip |
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| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
| abbrev Space := EuclideanSpace ℝ (Fin 3) |
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|
| def vec (x y z : ℝ) : Space := !₂[x, y, z] |
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| def vertical : Space := vec 0 0 1 |
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| def denominator (q : ℝ) : ℝ := Real.sqrt (2 * (1 + q ^ 2)) |
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| def normal (q : ℝ) : Space := |
| vec ((1 + q) / denominator q) ((1 - q) / denominator q) 0 |
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| def axis (q : ℝ) : Space := |
| vec ((1 - q) / denominator q) (-(1 + q) / denominator q) 0 |
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| def rectangle (q A l r : ℝ) : Set Space := |
| {z | ∃ s ∈ Icc (-A) A, ∃ t ∈ Icc l r, z = s • normal q + t • vertical} |
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| def cylinder (q A l r : ℝ) : Set Space := |
| {z | ∃ b ∈ rectangle q A l r, ∃ s : ℝ, z = b + s • axis q} |
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| def area (S : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2 S).toReal |
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| def strip (β M a b : ℝ) (W q : ℝ → ℝ) : Set Space := |
| {z | ∃ x y t : ℝ, z = vec (β*x) (β*y) t ∧ t ∈ Icc a b ∧ |
| |x| ≤ M ∧ |y| ≤ M ∧ |(1+q t)*x+(1-q t)*y| ≤ W t} |
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| def IsHorizontalRectangle (C : RuledApproximation.Cylinder) : Prop := |
| ∃ q A l r : ℝ, 0 < A ∧ l < r ∧ C.axis = axis q ∧ |
| ((fun z : ((Submodule.span ℝ ({C.axis} : Set Space))ᗮ) => (z : Space)) '' C.base) = rectangle q A l r |
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| def IsRectangle (B : Set Space) : Prop := |
| ∃ c u v : Space, inner ℝ u v = 0 ∧ u ≠ 0 ∧ v ≠ 0 ∧ |
| B = {z | ∃ s ∈ Icc (0 : ℝ) 1, ∃ t ∈ Icc (0 : ℝ) 1, |
| z = c + s • u + t • v} |
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| def HasRectangularBase (C : RuledApproximation.Cylinder) : Prop := |
| IsRectangle ((fun z : ((Submodule.span ℝ ({C.axis} : Set Space))ᗮ) => |
| (z : Space)) '' C.base) |
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| end CylinderCovering.BeamStrip |
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| end |
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| end OAI |
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|
