| 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 |
|
|
