| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| namespace ShortenedBeam |
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| abbrev Space := EuclideanSpace ℝ (Fin 3) |
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| def point (x y t : ℝ) : Space := !₂[x / Real.sqrt 2, y / Real.sqrt 2, t] |
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| def tetrahedron : Set Space := |
| {z | ∃ x y t : ℝ, |t| ≤ 1 ∧ |x| ≤ 1-t ∧ |y| ≤ 1+t ∧ z = point x y t} |
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| def leftCoordinates (a x y t : ℝ) : Prop := |
| ∃ p h : ℝ, 0 ≤ p ∧ p ≤ 1 ∧ 0 ≤ h ∧ h ≤ 2-a ∧ |
| x = -1+h*p+t*(1-a*(2-a)*p^2) ∧ |
| y = -1+h*(1-p)+t*(-1+a*(2-a)*(1-p)^2) |
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| def beamMinus (a : ℝ) : Set Space := |
| {z | ∃ x y t : ℝ, |t| ≤ 1 ∧ leftCoordinates a x y t ∧ z = point x y t} |
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| def beamPlus (a : ℝ) : Set Space := |
| {z | ∃ x y t : ℝ, |t| ≤ 1 ∧ leftCoordinates a (-x) (-y) t ∧ z = point x y t} |
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| def velocity (a p : ℝ) : Space := |
| point (1-a*(2-a)*p^2) (-1+a*(2-a)*(1-p)^2) 0 |
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| def axisMinus (a p : ℝ) : Space := |
| ‖point 0 0 1 + velocity a p‖⁻¹ • (point 0 0 1 + velocity a p) |
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| def axisPlus (a p : ℝ) : Space := |
| ‖point 0 0 1 - velocity a p‖⁻¹ • (point 0 0 1 - velocity a p) |
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| def drift (a t : ℝ) : ℝ := 2*a*t-2*a^2*t^3 |
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| def width (a t : ℝ) : ℝ := |
| max (a*(1-2*t^2)+a^2*t^2*(1+(1-|t|)^2+t^2)+25*a^3) 0 |
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| def strip (a : ℝ) : Set Space := |
| {z | ∃ x y t : ℝ, |t| ≤ 1 ∧ |x| ≤ 1-t ∧ |y| ≤ 1+t ∧ |
| |x+y+drift a t*((x-y)/2)| ≤ width a t ∧ z = point x y t} |
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| end ShortenedBeam |
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| end |
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| end OAI |
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