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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 | import OAI.Geometry.CylinderCovering.Angular.EndpointCoverDefinitions
import OAI.Geometry.CylinderCovering.Convexity.SimilarityDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering
def endpointScaledBody : Set TriangularCovering.Space :=
(fun z => (1/TriangularCovering.H) • z) '' TriangularCovering.K
def AngularEndpointScaledStatement : Prop :=
Similarity.minimumArea endpointScaledBody = TriangularCovering.H/2 ∧
∀ k : ℕ, 14000 ≤ k →
∃ C D : Fin (2*(k^2+1)) → TriangularCovering.TriCylinder,
TriangularCovering.AngularFamily.PrescribedFamily C (1/(k:ℝ))
(fun j => AngularEndpoint.boundary (k^2) j)
(fun j => AngularEndpoint.boundary (k^2) (j+1))
(fun j => AngularEndpoint.alpha (k^2) j)
(fun j => AngularEndpoint.beta (k^2) j)
(fun j => AngularEndpoint.radius (1/(k:ℝ)) (1/1000) (k^2) j) ∧
(∀ i, (D i).axis = (C i).axis ∧
(D i).base = (fun z => (1/TriangularCovering.H) • z) '' (C i).base) ∧
(∀ z ∈ endpointScaledBody, ∃ i, z ∈ (D i).carrier) ∧
|TriangularCovering.totalArea D - (2/TriangularCovering.H)*
(1/4-(13/12000:ℝ)*(1/(k:ℝ))^2)| ≤
7*TriangularCovering.H*(1/(k:ℝ))^4 ∧
TriangularCovering.totalArea D < Similarity.minimumArea endpointScaledBody/2 ∧
(∑ i, TriangularCovering.area (D i).base /
Similarity.shadowArea endpointScaledBody (D i).axis) < 1/2
end CylinderCovering
end
end OAI
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