File size: 3,133 Bytes
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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Geometry.CylinderCovering.Angular.EndpointScaledDefinitions
import OAI.Geometry.CylinderCovering.Angular.EndpointCover
import OAI.Geometry.CylinderCovering.Convexity.TriangularSimilarity
namespace OAI
noncomputable section
open Set
open TriangularCovering
namespace AngularEndpoint
lemma reciprocal_H_square : (1/H)^2 = (1/2:ℝ) := by
rw [div_pow,H_sq]
norm_num
lemma endpointScaledBody_eq : CylinderCovering.endpointScaledBody =
CylinderCovering.Similarity.map (0:Space) (1/H)
(LinearIsometryEquiv.refl ℝ Space) '' K := by
change (fun z : Space => (1/H) • z) '' K = _
apply congrArg (fun f : Space → Space => f '' K)
funext z
simp [CylinderCovering.Similarity.map]
lemma endpointScaledBody_minimum :
CylinderCovering.Similarity.minimumArea CylinderCovering.endpointScaledBody = H/2 := by
rw [endpointScaledBody_eq,CylinderCovering.Similarity.minimumArea_map K 0
(one_div_ne_zero H_ne_zero),reciprocal_H_square]
change (1/2:ℝ)*A_min=H/2
rw [projection_minimum]
ring
theorem endpoint_scaled_cover : CylinderCovering.AngularEndpointScaledStatement := by
refine ⟨endpointScaledBody_minimum,?_⟩
intro k hk
obtain ⟨C,hpres,hcover,harea,hcost,hrel⟩ := endpoint_cover k hk
let D := fun i => (C i).similarity 0 (one_div_ne_zero H_ne_zero)
(LinearIsometryEquiv.refl ℝ Space)
have hDarea : totalArea D = (1/2:ℝ)*totalArea C := by
dsimp [D]
rw [similarity_totalArea,reciprocal_H_square]
have hDcover : ∀ z∈CylinderCovering.endpointScaledBody,∃ i,z∈(D i).carrier := by
rw [endpointScaledBody_eq]
exact similarity_covers C hcover 0 (one_div_ne_zero H_ne_zero)
(LinearIsometryEquiv.refl ℝ Space)
have hDrel : (∑ i,area (D i).base /
CylinderCovering.Similarity.shadowArea CylinderCovering.endpointScaledBody (D i).axis) =
relativeCost C := by
rw [endpointScaledBody_eq]
exact similarity_relativeCost C 0 (one_div_ne_zero H_ne_zero)
(LinearIsometryEquiv.refl ℝ Space)
refine ⟨C,D,hpres,?_,hDcover,?_,?_,by rw [hDrel]; exact hrel⟩
· intro i
constructor
· rfl
· change CylinderCovering.Similarity.transportedBase (C i).base (C i).axis
0 (1/H) (LinearIsometryEquiv.refl ℝ Space) = _
unfold CylinderCovering.Similarity.transportedBase
apply congrArg (fun f : Space → Space => f '' (C i).base)
funext z
simp [CylinderCovering.Similarity.baseMap,CylinderCovering.Similarity.project]
· have h2div : 2/H=H := by apply (div_eq_iff H_ne_zero).2; nlinarith only [H_sq]
have hid : totalArea D-(2/H)*(1/4-(13/12000:ℝ)*(1/(k:ℝ))^2) =
(H/2)*(totalArea C/H-(1/2-(13/6000:ℝ)*(1/(k:ℝ))^2)) := by
rw [hDarea,h2div]
field_simp [H_ne_zero]
ring
rw [hid,abs_mul,abs_of_pos (div_pos H_pos (by norm_num))]
calc
_ ≤ (H/2)*(14*(1/(k:ℝ))^4) :=
mul_le_mul_of_nonneg_left harea (div_nonneg H_pos.le (by norm_num))
_ = 7*H*(1/(k:ℝ))^4 := by ring
· rw [hDarea,endpointScaledBody_minimum]
rw [projection_minimum] at hcost
linarith only [hcost]
end AngularEndpoint
end
end OAI
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