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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 | import OAI.Geometry.TriangularCovering.Basic
namespace OAI
noncomputable section
open Set
namespace TriangularCovering.AngularFamily
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))
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))
end TriangularCovering.AngularFamily
end
end OAI
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