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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 48 49 50 | import OAI.Geometry.CylinderCovering.Beams.MainDefinitions
namespace OAI
noncomputable section
open Set
open scoped InnerProductSpace
namespace CylinderCovering.BeamStrip
theorem IsRectangle.isCompact {B : Set Space} (hB : IsRectangle B) : IsCompact B := by
obtain ⟨c,u,v,_,_,_,rfl⟩ := hB
have heq : {z | ∃ s ∈ Icc (0:ℝ) 1, ∃ t ∈ Icc (0:ℝ) 1,
z = c+s • u+t • v} =
(fun p : ℝ × ℝ => c+p.1 • u+p.2 • v) '' (Icc (0:ℝ) 1 ×ˢ Icc (0:ℝ) 1) := by
ext z
constructor
· rintro ⟨s,hs,t,ht,rfl⟩
exact ⟨(s,t),⟨hs,ht⟩,rfl⟩
· rintro ⟨⟨s,t⟩,⟨hs,ht⟩,rfl⟩
exact ⟨s,hs,t,ht,rfl⟩
rw [heq]
exact (isCompact_Icc.prod isCompact_Icc).image (by fun_prop)
theorem HasRectangularBase.isCompact {C : RuledApproximation.Cylinder}
(hC : HasRectangularBase C) : IsCompact C.base :=
Subtype.isCompact_iff.mpr (IsRectangle.isCompact hC)
end CylinderCovering.BeamStrip
namespace ShortenedBeam
theorem HasTriangularBase.isCompact {C : RuledApproximation.Cylinder}
(hC : HasTriangularBase C) : IsCompact C.base := by
obtain ⟨v,hv,hbase⟩ := hC
rw [Subtype.isCompact_iff,hbase]
exact (Set.finite_range v).isCompact_convexHull ℝ
theorem HasMixedBase.isCompact {C : RuledApproximation.Cylinder}
(hC : HasMixedBase C) : IsCompact C.base := by
rcases hC with hC | hC
· exact hC.isCompact
· exact hC.isCompact
end ShortenedBeam
end
end OAI
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