| import OAI.Geometry.TriangularCovering.Basic |
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|
| namespace OAI |
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|
| noncomputable section |
| open Set |
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|
| namespace TriangularCovering.AngularFamily |
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|
| def LowerHit (ε a b A B R x y t : ℝ) : Prop := |
| 0 ≤ t - ε*x*B ∧ t - ε*x*B ≤ R ∧ |
| a*(t - ε*x*B) ≤ y - ε*x*A ∧ y - ε*x*A ≤ b*(t - ε*x*B) |
|
|
| def UpperHit (ε a b A B R x y t : ℝ) : Prop := |
| 0 ≤ 1 - t + ε*y*B ∧ 1 - t + ε*y*B ≤ R ∧ |
| a*(1 - t + ε*y*B) ≤ x + ε*y*A ∧ x + ε*y*A ≤ b*(1 - t + ε*y*B) |
|
|
| def lowerDirection (ε A B : ℝ) : Space := !₂[1, ε*A, H*ε*B] |
| def upperDirection (ε A B : ℝ) : Space := !₂[-ε*A, 1, H*ε*B] |
|
|
| def lowerIntercept (a b R : ℝ) : Set Space := |
| convexHull ℝ (range (![point 0 0 0, point 0 (a*R) R, point 0 (b*R) R] : Fin 3 → Space)) |
|
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| def upperIntercept (a b R : ℝ) : Set Space := |
| convexHull ℝ (range (![point 0 0 1, point (a*R) 0 (1-R), point (b*R) 0 (1-R)] : Fin 3 → Space)) |
|
|
| def PrescribedLower (C : TriCylinder) (ε a b A B R : ℝ) : Prop := |
| C.axis = ‖lowerDirection ε A B‖⁻¹ • lowerDirection ε A B ∧ |
| C.base = projection C.axis '' lowerIntercept a b R |
|
|
| def PrescribedUpper (C : TriCylinder) (ε a b A B R : ℝ) : Prop := |
| C.axis = ‖upperDirection ε A B‖⁻¹ • upperDirection ε A B ∧ |
| C.base = projection C.axis '' upperIntercept a b R |
|
|
| def PrescribedFamily {n : ℕ} (C : Fin (2*n) → TriCylinder) |
| (ε : ℝ) (a b A B R : Fin n → ℝ) : Prop := |
| (∀ j : Fin n, PrescribedLower |
| (C (Fin.cast (two_mul n).symm (Fin.castAdd n j))) ε (a j) (b j) (A j) (B j) (R j)) ∧ |
| (∀ j : Fin n, PrescribedUpper |
| (C (Fin.cast (two_mul n).symm (Fin.natAdd n j))) ε (a j) (b j) (A j) (B j) (R j)) |
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|
| end TriangularCovering.AngularFamily |
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| end |
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| end OAI |
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|
